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issue 81
SEQUENCES (RE)DISCOVERED
We recently discovered website of the On-line Encyclopedia of Integer Sequences by N. J. A. Sloan, www.research.att.com/~njas/sequences, and discovered several of our favorite ones were not included in its 70,000+ entries. We submitted some and got them accepted. Rather mysterious-looking, aren't they? (Exercise your brain and see if you can spot the patterns before reading the generating rule.)
I. 0 1 7 2 5 8 8 3 11 5 9 9 8 9 9 4 9 12 14 7 7 11 12 10
II. 10 11 13 41 51 16 17 81 91 30 13 33 43 53 63 73
III. 1000 1000000000 1000000000000000000000000000 100 1 4 8 3 5
IV. 5 6 8 9 11 12 13 15 16 18 19 25 26 28 29
V. 1 3 4 5 6 7 8 9 10 11 13 14 15 16 17 18 19 30 31 33 34 35 36 37 38 39 40 41 43 44
VI. 1652100 31946 38760 49537526 732051 724298 36969 47723135 24375809
VII. 18 666 23994 679 31 1134 40842 1470330 681 33 1206 43434 1563642 43974
VIII. 1 2 3 4 7 10 14 17 20 21 22 23 24 27 40 41 42 43
IX. 0 3 6 8 9 30 33 36 38 39 60 63 68 69 80 83 86 89 90 93 96 98 99 300 303
X. 2 5 22 25 52 55 222 225 252 255 522 525 552 555 2222 2225 2252 2255
XI. 11 101 11 101 11 101 11 101 11 22 112 22 112 22 112 22 112 22 112 202 112 202 112
XII. 5 2 5 2 5 2 5 2 5 2 4 2 4 2 4 2 4 2 4 3 2 3 2 3 2 3 2 3 2 3 4 2 4 2 4 2 4 2 4 3 2
XIII. 10000 10001 10002 10003 10004 10005 10006 10007 10008 10009 10010
XIV. 3 141 5926535 89793238 4626433 8327950 2 88 4 1971 6939975 10 5 820
XV. 11 22 33 44 55 66 77 88 99 110 111 112 113 114 115 116 117 118 119 220 221 222
XVI. 0 1 2 4 6 8 9 10 11 12 14 16 18 19 20 21 22 23 24 26 28 29 40 41 42 44 46 48
XVII. 3 14 15 92 6535 8979 323846 26433832 795028 841971 69399375
XVIII. 5 3 6 2 3 7 2 3 5 4 2 4 8 3 2 3 6 3 2 3
XIX. 1 3 6 7 8 9 10 13 16 17 18 19 30 31 33 36 37 38 39 61 63 66 67 68 69 70 71 73
XX. 4 2 3 6 2 3 7 2 3 5 4 2 4 8 3 2 3 6 5 2 3 5
XXI. 3 5 6 7 9 10 11 12 13 15 16 17 19 20 23 25 26 27 29 30 33 35 36 37 39 50 53
XXII. 1 2 4 8 14 18 21 22 24 28 31 32 34 38 40 41 42 43 44 45 46 47 48 49 51 52
XXIII. 101,000 1010,000 10100,000 101,000,000 1010,000,000 10100,000,000 101,000,000,000 1010,000,000,000
XXIV. 1728 20736 248832 2985984 35831808 429981696 5159780352
XXV. 0 1 4 8 9 10 16 25 36 40 49 64 80 81 100 121 125 144 160 169 196 216 225 250
XXVI. 1 2 5 6 7 8 9 10 11 12 15 16 17 18 19 20 29 50 51 52 55 56 57 58 59 60 61 62 65 66 67 68 69 70 71 72 75 76 77 78 79 80 81 82 85 86 87 88 89 90 91 92 95 96 97 98 99 1000000000
XXVII. 0 1 2 4 5 6 8 10 24 26 40 46 64 84 200 206 600 5000
XXVIII. 120 121 123 124 125 126 127 128 129 320 321 323 324 325 326 327 328
XXIX. 1, 16, 1.26630716... x 103,638,334,640,024
XXX. 0 2 3 4 8 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
XXXI. 1 18 24 2753 59 62 95 98 126 132 135
XXXII. 3 141 592 653 58979 3238462 6433832 7950288 419716939 9375105820
XXXIII. 12 21 22 2324 25 26 27 28 29 32 42 52 62 72 82 92 102 112 122 132 142
XXXIV 142857 5882352941176470 526315789473684210 4347826086956521739130 3448275862068965517241379310
2127659574468085106382978723404255319148936170
XXXV. 1 5 10 25 40 63 84 110 135 159 192 230 265 294 330 366 397 434 455 483
XXXVI. 5 3 6 2 3 7 2 3 5 4 2 4 8 3 2 3 6 3 2 3 5 3 2 3 6 5 2 3 5 5 2 3 5 3 2 3 7 5 2 3 6 4 2 3 5 4 2 3 5 2 2 3 8 4 23 7 5 2 3 5 2 3 2 3 9 5 2
XXXVII. 4 2 3 6 2 3 7 2 3 5 4 24 8 3 2 3 6 5 2 3 5 3 2 3 6 3 2 3 5 5 2 3 5 5 2 3 7 3 2 3 6 5 2 3 5 4 2 3 5 4 2 3 8 3 2 3 7 5 2 3 5 2 3 2 3
XXXVIII. 11 12 15 24 36 111 112 115 128 132 135 144 175 212 216 224 312 315 384 432 612 624 666 672 735 816 1111 1112 1113 1115 1116
XXXIX. 1 265,536
XXXX. 2 3 4 5 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 32 33 34 35 37 38 39 40
XXXXI. 11318 15216 10799546 129618 125258 14118 10211981 2839691 282506
XXXXII. 9 252 6813 265 22 603 16290 439839 267 24 657 17748 479205 17761 670 18099 488682 13194423 17763 672 18153 490140 13233789 490153 18166 490491
XXXXIII. 1 9 31 36 98 107 156 164 210 221 266 312 358 365 405 415 460 467 509 519 548 556 564 566 571 577 587
XXXXIV. 3 14 15 96 535 897 933 8466 43383 79502 88419 716939 937510
XXXXV. 3 14 19 23 89 793 2384 2433 8327 9028 84197 193993 710820 974944
XXXXVI. 3 11 59 535 8979 33833 83795 88197 193993 751589 795937 818899
XXXXVII. 3 14 19 26 89 92 846 2648 2902 8841 9169 9910 82094 944920 816406
XXXXVIII. 4 26 82 84 626 4820 28846 10820 44420 86406 286208 862804
XXXXIX. 3 11 59 535 897 933 8338 37950 197169 399375 1058097 9593071
L. 2 10 30 68 130 222 350 520 738 2 4 12 32 70 132 224 352 522 740 10 12 20 40
LI. 1 2 5 8 9 10 11 12 15 18 19 20 21 22 25 28 29
LII. 1 2 2.799137013...x10507(1010)142,581
LIII. 0 1 4 7 64 100 101 104 107 343 401 404 407 700 701 704 707 764
LIV. 1 4 11 16 19 29 33 42 56 70 71 74 77 87 105 109 121 128 132 142 151 161 166
LV. 3 6 27 12 15 216 21 24 729 30 33 9 39 42 126 48 51 513 57 60 9 66 69 72 75 78
LVI. 4 4 3 5 4 3 5 4 3 4 5 4 4 4 3
LVII. 1 5 6 7 9 11 1000000 1000001 1000005 1000006 1000007 1000009 1000011
LVIII. 2 9 16 23 30 6 -142 -600 -1678 -3841 -7740 -14243 -24466
LIX. 1 2 5 6 7 8 9 10 11 12 15 16 17 18 19 20 21 22 25 26 27 28 29 50 51 55 56 57 58
LX. 2 40 42 50 52 90 92 200 240 242 250 252 290 292 2000000 2000002
LXI. 0 8 69 96 609 689 906 986 6009 6699 6889 6969 9006 9696 9886 9966
LXII. 5 6 7 9 11 500 505 506 507 509 511 600 605 606 607 609 611 700 705 706
I. The most interesting one is perhaps that involved with the 65-year-old unsolved 3n+1 problem. All integers seem to be reducible to 1, but it hasn't been proven yet. modified Collatz: [Collatz or hailstone delay sequence, A008577, modified to allow application of 3x+1 operation on even numbers for reduced delay, D(n)] The sequence, is generated by applying the HOTPO operator on the integers until each is reduced to 1and listing the number of steps needed. The number of steps for 9, for example, is 119 in the unmodified Collatz. It's also called a hailstone sequence because the HOTPO operator makes the integer go up and down apparently at random like a hailstone in a hailstorm.
II. abntu: [pronounced “ab-`n-too”, “Word Weirdness”, Mpossibilities 66:5, Feb. 1998, Bantu alphabeticalized, A072809]
III. alphabetical numeration: [“Alphabetical Numeration”, Puzzle-M (Feb. 1987), least number containing each letter of alphabet, using base 36 values for jillion and kazillion, related to A053433]
IV. Ariel: [antonym of Caliban, with a, c, i or l] complement to Caliban
V. Bantu: [“Word Weirdness”, Mpossibilities 66:5, (Feb. 1998), from “ban two”]
VI. base-36: [number names converted from base 36 in which 10 = A, 11 = B, ..., Z = 35, A072922]
VII. base-36 Roman numerals: [Roman numerals converted from base 36]
VIII. Caliban: [without a, c, i, l, A072958]
IX. curvaceous: [“Three Boxes”, Puzzle-M (Apr. 1987), with digits having curved lines, A072960]
X. curvilinear: [“Three Boxes”, Puzzle-M (Apr. 1987), with digits having both curved and linear lines, A072961]
XI. DENEAT: [“Blackholing”, Mpossibilities 69:2, (Jan. 1999), “digits -- even, not even and total”, Michael Ecker, New Scientist Dec. 1992, A073053]
XII. deneaticity: [“Blackholing”, Mpossibilities 69:2, (Jan. 1999), Michael Ecker, “Caution: Black Holes at Work”, New Scientist Dec. 1992, “digits -- even, not even and total”, A 073054]
XIII. diamond: [Robert E. Smith, number with n2+(n-1)2 digits arrangable in diamond, 104 to 105-1, then 1012 to 1013-1, etc.]
XIV. diced pi: [pi divided into integers in (and alternately outside) cube digits, 0, 1 or 8, with initial 0 understood when number ends in cube]
XV. double-header: [number with first two digits identical]
XVI. ellav: [“Word Weirdness”, Mpossibilities 66:6 (Feb. 1998), ananym of valle, without 3, 5, 7]
XVII. evenly sliced pi : [pi sliced into ever increasing even-digited integers]
XVIII. five's: [“Newies”, Mpossibilities 64:3 (Mar. 1997), A072424, from the generating sentence: "Five's the number of letters in the first word of this sentence, three in the second, six in the third, two in the fourth, three in the fifth ..."]
XIX. flawless: [without a, f, l, w]
XX. four-ises: [“Newies”, Mpossibilities 64:3 (Mar. 1997) On-line Encyclopedia of Integer Sequences by N. J. A. Sloan, A072425, f rom the generating sentence: "Four is the number of letters in the first word of this sentence, two in the second, three in the third, six..."]
XXI. godless: [finite sequence without d, g, o]
XXII. godly: [with d, g, o]
XXIII. great googol: [Mpossibilities 74:6 Aug. 2000, from analogy, gross:great gross::googol:?]
XXIV. great gross [12n+2]
XXV. Haken: [decimal multiples or squares or cubes
XXVI. harmless: [without a, h, m, r]
XXVII. heterogram: [Susan Thorp, with no letters repeated, A059916]
XXVIII. intwo: [number with 2 in interior, but on neither end]
XXIX. JGE: [Joyce Generalized Exponential, J(n1, n2, n3, ..., nm-2, nm-1, nm) = J( n1, J(n1-1, n2, n3, ..., nm-2, nm-1, nm), nm), J(m, n, p, q) = Gm(n, p, q) = Gm-1(n, G(n, p, q), p), J(n, ..., n)]
XXX. nabrut: [ ananym of turban]
XXXI. n-ests: [“Newies”, Mpossibilities 64:3 (Mar. 1997) On-line Encyclopedia of Integer Sequences by N. J. A. Sloan, A072422 (July 31, 2002), from the generating sentence, “N est prima littera in hic sententiam, doudevicesima littera in hic sententiam, quarta vicesima littera in hic sententiam, septima vicesima littera in hic sententiam, tertia quinquagentesima littera in hic sententiam ...."] XXXII. oddly sliced pi: [pi sliced into ever increasing odd-digited integers]
XXXIII. ontwo: [“Word Weirdness”, Mpossibilities 66:5 (Feb. 1998), with 2 on either end
XXXIV. persistent: [“Overbyte”, Mpossibilities 62:2 (Apr. 1996), Believe It or Not by Robert Ripley (1929), number whose sequence of digits persists when multiplied, [10p-1/p], at least in the first few cases the rounded quotient of ten-to-a-prime-minus-one over the so-called repetend prime, A006883]
XXXV. p-ests: [“Newies”, Mpossibilities 64:3 (Mar. 1997), On-line Encyclopedia of Integer Sequences by N. J. A. Sloan, A072421 (July 31, 2002), from the generating sentence, "P est prima praeterea quinta praeterea decima praeterea quinta vicesima praeterea quadragesima praeterea tertia sexagesima praeterea quarta octogesima praeterea decima centesima ... littera in hic sententiam."] XXXVI. pfive's: [five's that is also palindromic]
XXXVII. pfour-is: [four-is that is also palindromic]
XXXVIII. podd: [number whose product-of-digits divides the number, notable in that a(666) = 666]
XXXIX. RGE tactorial: [“New Ways to Get High”, Mpossibilities 70:4 Mar. 1999, Recursive Generalized Exponential, !n!G(!n!, !n!, !n!)]
XXXX. s-ain't: [“When the S-ain'ts ...”, Mpossibilities 61:1-2 (Feb 1996), A072886, numbers generated like the Aronson series from a generating sentence, "S-ain't the second, third, fourth, fifth . . . letter of this sentence.".]
XXXXI. Sallows': [integers in Lee Sallows' base-27 system where space = 0, A = 1, B = 2, etc., A072959]
XXXXII. Sallows' Roman numerals: [Roman numerals evaluated as if in Sallows' base 27]
XXXXIII. s-inner: [“When the S-ain'ts ...”, Mpossibilities 61:1-2 (Feb 1996), A072887, numbers generated like the Aronson series from a generating sentence, "S-ain't the second, third, fourth, fifth . . . letter of this sentence.”]
XXXXIV. sliced Bantu pi: [pi without 2, sliced into ever increasing integers]
XXXXV. sliced Caliban pi: [pi without 5, 6 sliced into ever increasing intergers] XXXXVI. sliced eban pi: [pi without digits with e, (0, 2, 4, 6), sliced to form ever increasing integers,
XXXXVII. sliced ellav pi: [Mpossibilities 66:6, ananym of valle, term from Paguingue, without 3, 5, 7, sliced into ever increasing integers]
XXXXVIII. sliced even pi: [Mpossibilities 64:3, pi without odd digits; sliced into ever incresing integers]
XXXXIX. sliced evenly-evenly pi: [Mpossibilities 64:2, 80, pi without 2, 4 or 8 digits; sliced to form ever increasing integers]
L. SODAC: [ “sum of digits and cubes]
LI. suburban: [without b, r, s, u, A072955]
LII. tactorial: [“New Ways to Get High”, Mpossibilities 70:3, Mar. 1999, !n! = !(n!)]
LIII. Taliban: [without letters a, i, l, t, A072954]
LVI. t-ests: [A072423, from the generating sentence: "T est prima et quarta et undecima et sexima decima et nona decima et nona vicesima ... littera in hic sententiam."]
LV. TOSCOD: [“triple or sum cubes of digits”]
LVI. toscodicity: [On-line Encyclopedia of Integer Sequences by N. J. A. Sloan, A72420, number of operations of TOSCOD to tranform n to 153]
LVII. turban: [without letters r, t, u, A072956]
LVIII. unexpected: [Mpossibilities 36:4, a(n) = 7n - 5 - 31(n-1)(n-2)(n-3)(n-4)(n-5)/5!, alternative answer to “2 9 16 23 30 6 ?”, said to be best answered with 7n + 2 (mod 31) from calendar]
LIX. urban: [without r, u, A072957]
LXXIII. useless: [without e, s, u]
LXXIV. vertical palindromes: [numbers that read the same up-side-down]
LXXV. worthless: [without h, o, r, t, w]
P A R A N O R M A L I N S I D E R {Aug. 17, 2002)
Concluded Dr. Wise, "It would be easy to conclude that these elderly people had entered the early stages of senile dementia. However, not all experts agree that brain functions degenerate to this degree as we age. An alternate explanation by some in the paranormal community is that frail old people are more vulnerable to attack by unseen entities that gain mischievous pleasure from altering their beliefs and behavior.”
Live On TV: The Loch Ness Monster! You might just catch a live glimpse of Nessie when you visit the camvista.com web site. That's because the company has a camera mounted on Deepscan, the research vessel of the Loch Ness Project. Headed by Adrian Shine, the project offers live pictures from the ship's deck as well as from the vantage point of an underwater camera that is periodically lowered into the depths.
NEWS OF THE WEIRD
Mary Palmieri, Enfield, Conn., is now homeless after she allowed pagan “friends” to perform a ritual in her house to "burn her troubles away." The witchcraft ritual involved burning a piece of paper with Mary's problems written on it. The flames got out of control and set fire to the house. Mary's bedroom was gutted and the house suffered extensive smoke and water damage. Mary says next time she will talk to her priest instead.
"A woman, who told Roswell Police she had been on another planet for three years, reported a robbery Friday. She said a known person had taken the upper plate of her dentures valued at $800, silverware in a wooden box valued at $1,000, and various jewelry worth $1,000. She said she hadn't actually seen the named suspect take the items, but he 'moves so swift you can't see him.'" [Roswell Daily Record, 5-29-01]
In preparation for the founding meeting of a new political group (the Anambra Peoples' Forum) in Lagos, Nigeria, in March, officials concerned about being rained out hired a professional rain doctor, Mr. Chief Nothing Pass God, for about $47 (and a bottle of gin) to keep the skies clear. Before the doctor was finished with his incantations, a rare March downpour completely washed out the event. Said the Chief, "I have not failed. What caused the disappointment was that (this job) came unexpected(ly)" and that he had not had sufficient time to prepare. [South African Press Association-Agence France-Presse, 3-18-02]
BIZARRE NEWS
When Kathleen Healy, Rosedale, Minn., entered the Marshall Field's dressing room, she found a money clip full of cash. Though she could see a $50 and $100 bill, she didn't count it all because "it wasn't my money," she said. She then gave the wad to the clerk and refused any reward. The sales clerk "screamed and said 'Oh, a customer's been looking all over for this.'" The frantic customer burst into tears of relief when they returned the cash to her, but Healy still refused a reward. The sales clerk insisted she take a box of Marshall Field's signature chocolates, Frango Mints. When Healy opened the box, a note was enclosed indicating she had won $10,000. As part of their "Win a Mint" game. "I've never won anything before. I've never won a toaster," the ecstatic winner said.
COMING NEXT ISSUE ---
THE TRUTH (MORE OR LESS) ABOUT CROP CIRCLES
perfects: 6 28 496 8128 33550336 8589869056 137438691328
2305843008139952128 2658455991569831744654692615953842176
Tchoukaillon (or Mancala, or Kalahari): [keep 1st number, drop every 2nd, keep 1st, drop every 3rd, keep 1st, drop every 4th, etc., A007952 “Kalahari and the Sequence” by D. Betten, "Sloane No. 377", Annals Discrete Math., 37:51-58, 1988.] 0 1 3 5 9 11 17 21 29 33 41 47 57 59 77 81 101 107 117 131 149 153 173 191 209 213 239 257 273 281 321 329 359 371 401 417 441 453 497 509 539 569 611 621 647 671 717 731 779 801 839 869 917 929 989 1001 1053 1067
RATS: [Reverse Add Then Sort the digits applied to previous term, starting with 1. “Conway's RATS and other reversals” by R. K. Guy, Amer. Math. Monthly, 96:425-8(1989)] 1 2 4 8 16 77 145 668 1345 6677 13444 55778 133345 666677 1333444 5567777 12333445 66666677 133333444 556667777 1233334444 5566667777 12333334444 55666667777 123333334444 556666667777 1233333334444
It is conjectured that no matter what the starting term is, repeatedly applying RATS leads either to this sequence or into a cycle of finite length, such as those in A066710 and uences/eisA.cgi?Anum=A066711" A066711.
possible chess games after n moves: [A004000, Ken Thompson] 1, 20, 400, 8902, 197281, 4865617
ebans: [e-lipogram of integers, sieve that removes all integer names with an e, A006933] 2 4 6 30 32 34 36 40 42 44 46 50 52 54 56 60 62 64 66 2000 2002 2004 2006 2030 2032 2034 2036 2040 2042 2044 2046 2050 2052 2054 2056 2062 2064 2066 4000 4002 4004 4006 4030 4032 4034 4040 4042 4044 4046
even sieve: [A056533 start with natural numbers, remove every 2nd term, remove every 4th term from what remains, remove every 6th term from what remains, etc.]
1 3 5 9 11 17 19 25 27 35 37 43 51 57 59 69 75 83 85 97 101 113 117 129 131 147 153 161 163 181 185 195 203 211 219 233 243 257 259 273 275 291 307 315 321 339 341 357 369 387 389 401 417 425 437 453 465
even lucky numbers: [A045954, generated by a sieve process like that for lucky numbers but starting with even numbers] 2 4 6 10 12 18 20 22 26 34 36 42 44 50 52 54 58 68 70 76 84 90 98 100 102 108 114 116 118 130 132 138 140 148 150 164 170 172 178 182 186 196 198 212 214 218 228 230 234 244 246 260 262 268 278 282 290 298 300 308
lucky numbers: [A005589] 1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73 75 79 87 93 99 105 111 115 127 129 133 135 141 151 ,159 163 169 171 189 193 195 201 205 211 219 223 231 235 237 241 259 261 267 273 283 285 289 297 303
Smiths: [A006753, sum of digits = sum of digits of prime factors(counted with multiplicity] 4 22 27 58 85 94 121 166 202 265 274 319 346 355 378 382 391 438 454 483 517 526 535 562 576 588 627 634 636 645 648 654 663 666 690 706 728 729 762 778 825 852 861 895 913 915 922 958 985 1086
palindromic primes: [A002385 Recreations in the Theory of Numbers by A. H. Beiler, Dover, NY, 1964, p.228]
2 3 5 7 11 101 131 151 181 191 313 353 373 383 727 757 787 797 919 929 10301 10501 10601 11311 11411 12421 12721 12821 13331 13831 13931 14341 14741 15451 15551 16061 16361 16561 16661 17471 17971 18181
emirps: [prime semordnilaps or palindromic pairs, A006567, The Magic Numbers of Dr Matrix by Martin Gardner (Prometheus, Buffalo, NY, 1985, p. 230)] 13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389 701 709 733 739 743 751 761 769 907 937 941 953 967 971 983 991 1009 1021 1031 1033 1061 1069 1091 1097 1103 1109 1151 1153 1181 1193
smallest number requiring n syllables in English: [A002810, Problems Drive, Eureka, 37:8-11, 33 (1974)] 1 7 11 27 77 107 111 127 177 777 1127 1177 1777 7777 11777 27777 77777 107777
111777 127777 177777 777777 1127777 1177777 1777777 7777777
exp = exponential
_ = subscription
^ = superscription
a(n) = nth term
continued fraction in example line .xxxxxxx...
decimal expansion of .xyz... = x, y, z, ...
triangle, square, etc (tabl) row-by-row
convert(n, base m)
fractions = (frac) as double, crossrefenced sequences
tabf funny-shaped table
Name: .
(Formerly M2277)
Sequence: 4,3,3,5,4,4,3,5,5,4,3,6,6,8,8,7,7,9,8,8,6,9,9,11,10,10,9,11,
11,10,6,9,9,11,10,10,9,11,11,10,6,9,9,11,10,10,9,11,11,10,5,
8,8,10,9,9,8,10,10,9,5,8,8,10,9,9,8,10,10,9,7,10,10,12,11,
11,10,12,12,11,6,9,9,11
A051785
Sequence: 2,3,4,6,4,3,3,4,3,3,4,5,6,7,6,5,6,6,5,4,7,8,9,11,9,8,8,9,8,
6,8,9,10,12,10,9,9,10,9,8,10,11,12,14,12,11,11,12,11,9,11,
12,13,15,13,12,12,13,12,8,10,11,12,14,12,11,11,12,11,7,9,10,
11,13,11,10,10,11,10,8,10,11
Name: Number of letters in n-th Catalan number.
References J. Gili, Catalan Grammar, Dolphin. Oxford 1993, p. 39.
Amer. Math. Monthly 103 (1996), 538 and 577.
R. P. Stanley, Enumerative Combinatorics, vol 2, problem 6.24.
Example: In Catalan: un, dos, tres, quatre, cinc, sis, set, vuit, nou, deu, onze,dotze, tretze, catorze, quinze, setze, disset, divuit, dinou, vint,
...
Fibonacci numbers whose sum of digits sets a new record. A068500M. Gardner, Penrose Tiles to Trapdoor Ciphers. Freeman, NY, 1989, p.300]
1 2 3 5 8 55 89 987 28657 196418 1346269 3524578 5702887
39088169 267914296 4807526976 7778742049 139583862445
591286729879 1304969544928657 5527939700884757
99194853094755497
References
Example: a(8)=987 and 9+8+7=24 and sum of digits of any fibonacci numbers <987 is also less than 24 .
Keywords: nice,nonn
Offset: 1
A039672
ALLDIFFS all differences a(j)-a(i), j>i
# ANDCONV AND-convolution, SUM a(k).AND.a(n-k)
# BINOMIAL from 1st diag of diff table to top row
# BINOMIALi inverse: from sequence to leading diag of diff table
# BIN1 a variant of BINOMIAL used by Zagier
# BISECT(a,0), BISECT(a,1) bisect a sequence
# BOUS boustrophedron transform
# BOUS2 boustrophedron transform (official boust. transform)
# BOUS2i inv. boust transform (official boust. transform)
# CHAR characteristic function of sequence
# COMP complement of sequence
# COMPl complement of sequence (long version)
# COMPOSE compose two sequences
# COMPSQRT functional square root
# CONTINUANT continuant transform, cf. Concrete Math. p. 301
# CONTINUANTi inverse continuant transform (not always integral)
# CONV simple convolution, expand A(x)*B(x)
# CONVi square root of convolution, not always integral
# DECIMATE(a,k,0), DECIMATE(a,k,1), ... decimate a sequence
# DIFF first difference
# DIGREV reverse digits (use digrev for single numbers)
# DIGSUM sum of digits (use digsum for single numbers)
# DIRICHLET Dirichlet convolution of two sequences
# EULER Euler Xfm
# EULERi inverse Euler Xfm
# EXP egf of b = exp (egf of a)
# EXPCONV exponential convolution, expand E1(x)*E2(x)
# EXTRACT extract subsequence
# GCDCONV GCD-convolution, SUM a(k).gcd.a(n-k)
# HANKEL Hankel transform
# HEAP Given a, start with heap of 1, add largest entry in a <= heap to heap; b gives successive sizes of heaps
# INVERT a's from b's in 1+SUM a_i x^i = 1/(1-SUM b_i x^i)
# INVERTi inverse, b's from a's
# LCMCONV LCM-convolution, SUM a(k).lcm.a(n-k)
# LEFT shift left
# lHANKEL little Hankel transform
# LISTTOLISTDIV divides nth term by n!
# LISTTOLISTMULT multiplies nth term by n!
# LOG egf of b = log (egf of a)
# MASKCONV mask-convolution
# MASKTRANS mask-convolution with all-1's sequence
# MASKTRANSi inverse
# MOBIUS Mobius (or Lambert) transform
# MOBIUSi inverse Mobius (or sum of divisors) transform
# MONO sort, discard duplicates
# MONO2 sort, discard duplicates, but only if difft from orig.
# M2 multiply all except 1st term by 2
# M2i divide all except 1st term by 2
# NEGATE negate all except 1st term
# ORCONV OR-convolution, SUM a(k).OR.a(n-k)
# PARTITION partition Xfm (without repetition)
# PARTITIONi inverse partition Xfm (without repetition)
# POLY get polynomial for seq from difference table
# POWERS extracts powers of x in f through degree n
# PRODS sequence of partial products
# PSUM partial sums
# PSUMSIGN alternating sign partial sums
# REVERT reversion
# REVEGF reversion of e.g.f.
# RIGHT shift right, prepending a 1
# SERIESTOLISTDIV divides nth term by n!
# SERIESTOLISTMULT multiplies nth term by n!
# SERIESTOSERIESDIV divides nth term by n!
# SERIESTOSERIESMULT multiplies nth term by n!
# SERIES2 series expansion of function of 2 variables
# SERIES2TOLIST converts output of SERIES2 to a list, order 00,10,01,20,11,02,...
# SERIES2TOLISTMULT ditto, multiplying coefft of x^i*y^j by i!*j!
# SERIES2HTOLIST converts output of SERIES2 to a list, order 00,10,11,20,21,22,...
# SERIES2HTOLISTMULT ditto, multiplying coefft of x^i*y^j by i!*j!
# SHADOW see Lorenz Halbeisen and Norbert Hungerbuehler reference in sequence database under A000522
# STIRLING Stirling Xfm (of 2nd kind)
# STIRLINGi inverse Stirling Xfm (equivalently, Stirling transform of 1st kind)
# STIRB Stirling-Bernoulli transform
# SUPPORT positions where list is nonzero
# TRISECT(a,0), .. TRISECT(a,2) trisect a sequence
# WEIGH b from a in 1+SUM b_n x^n=PI (1+x^a_n)
# XORCONV XOR-convolution, SUM a(k).XOR.a(n-k)
# others:
# delta delta(f,n,k) = kth term of nth difference of f
# did did it divide?
# dids did it divide (with sign)?
# digrev reverse digits
# digsort sort digits
# digsum sum of digits
# dim did it mask?
# Etrans Euler Xfm (again)
# mex minimum excluded number
# nimsum nim sum
# ptrans partition Xfm (with repetition)
# traptrans inverse partition Xfm (with repetition)
# weighout b from a in 1+SUM b_n x^n=PI (1+x^n)^a_n
# weighouti a from b in 1+SUM b_n x^n=PI (1+x^n)^a_n
# weighini a from b in 1+SUM b_n x^n=PI (1+x^a_n)
# weigh2out b from a in 1+SUM b_n x^n=PI (x^-n+1+x^n)^a_n
# weigh2outi a from b in 1+SUM b_n x^n=PI (x^-n+1+x^n)^a_n
# weigh2in b from a in 1+SUM b_n x^n=PI (x^-a_n+1+x^a_n)
# weigh2ini a from b in 1+SUM b_n x^n=PI (x^-a_n+1+x^a_n)
# eultrans2 2nd Euler trans from paper by Donaghey and Shapiro
# but this is same as BINOMIALi
# pairtrans b(n)=a(n)+a(n-1)
# pairtransi a(n)=b(n)-b(n-1)+b(n-2)-...
# wt Compute weight or number of 1's in binary expansion of n:
# ANDnos logical AND of two numbers using their binary expansions
# ORnos logical OR of two numbers using their binary expansions
# XORnos logical XOR of two numbers using their binary expansions
# deriv derivative of a numbers using its binary expansion
ID Number: A005228 (Formerly M2629)
Sequence: 1,3,7,12,18,26,35,45,56,69,83,98,114,131,150,170,191,213,
236,260,285,312,340,369,399,430,462,495,529,565,602,640,679,
719,760,802,845,889,935,982,1030,1079,1129,1180,1232,1285,
1339,1394,1451,1509,1568,1628,1689
Name: Sequence and first differences (A030124) include all numbers
References D. Hofstadter, "Goedel, Escher, Bach", p. 73.
N. J. A. Sloane, My favorite integer sequences (xt" Abtract, pdf, ps), in Sequences and their Applications (Proceedings of SETA '98), C. Ding, T. Helleseth and H. Niederreiter (editors), Springer-Verlag, London, 1999, pp. 103-130.
E. W. Weisstein, Link to a section of The World of Mathematics.
om/~njas/sequences/Sindx_Go.html" \l "GEB" Index entries for sequences from "Goedel, Escher, Bach"
Formula: a(n) = a(n-1) + c(n-1) for n >= 2, where a(1)=1, a( ) increasing, c( ) =
complement to a( ). (c is the sequence //www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=030124" A030124).
Sequence reads 1 3 7 12 18 26 35 45..., differences are 2 4 5, 6, 8, 9, 10 ..., and the point is every number not in the sequence itself appears among the differences. This property (together with the fact that the sequence is increasing) defines the sequence!
binary sieve: [A007950, delete every 2nd number, then every 4th, 8th, etc. "Only Problems, not Solutions!" by F. Smarandache, Xiquan Publ., Phoenix-Chicago, 1993]
] 1,3,5,9,11,13,17,21,25,27,29,33,35,37,43,49,51,53,57,59,65,
67,69,73,75,77,81,85,89,91,97,101,107,109,113,115,117,121,
123,129,131,133,137,139,145,149
Sequence: 1,2,4,5,7,10,11,13,16,19,20,23,25,28,29,32,37,38,40,41,49,
50,52,56,58,59,61,65,68,74,76,77,82,83,86,88,91,97,101,103,
104,106,115,118,121,122,124,130,131,133,136,137,149,151,154,
155,158,163,164,166,173,175
Name: Fibonacci-lucky numbers: generated by a sieve process with Fibonacci
rule.
Comments: Start with 1 2 3 4 5 6 7 8 9 10 11 12 13...; sum of 1st and 2nd terms = 3; strike
out every 3rd term: 1 2 4 5 7 8 10 11 13...; sum of 2nd and 3rd terms = 6; strike
out every 6th term: 1 2 4 5 7 10 11 13...; sum of 3rd and 4th terms = 9; strike
out every 9th number; etc.
Math'ca: Fold[ Delete[ #1,Thread[ List[ Rest[ Range[ 0,Length[ #1 ],#1[ [ #2 ]
]+#1[ [ #2+1 ] ] ] ] ] ] ]&, Range[ 200 ], Range[ 20 ] ]
Keywords: nice,nonn
Here is Sequence A006577 (this will take a moment):
ID Number: A006577 (Formerly M4323)
Sequence: 0, 1, 7, 2, 5, 8, 16, 3, 19, 6,
14, 9, 9, 17, 17, 4, 12, 20, 20, 7,
7,15, 15,10, 23,10,111 18, 18,18,
106, 5, 26,13, 13,21, 21, 21, 34, 8,
109, 8, 29,16, 16,16,104, 11, 24, 24,
24,11, 11,112,112,19,32, 19, 32, 19,
19,107,107,6, 27,27,27, 14, 14, 14,
102,22
0,1,7,2,5,8,16,3,11, 6,
14, 9, 9,17,
Name: Number of halving and tripling steps to reach 1 in modified`3x+1' problem.
References R. K. Guy, Unsolved Problems in Number Theory, E16.
Links: J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Geometry.net, _Conjectures/Collatz_Problem.htm" Links on Collatz Problem
Mathematical BBS, Biblography on Collatz Sequence
E. W. Weisstein, rld.wolfram.com/CollatzProblem.html" Collatz Problem
G. Villemin's Almanach of Numbers, Cycle of Syracuse
Example: a(5)=5 because the trajectory of 5 is (5,16,8,4,2,1).
See also: Cf. esearch.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=008908" A008908.
Keywords: nonn,nice,easy
Offset: 1
Author(s): njas, R. W. Gosper
Extension: More terms from Larry Reeves (larryr@acm.org), Apr 27 2001
www.research.att.com/cgi/as/njas/sequences
3 1415 9 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432
66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
triple dactorial: [(n!!!)!!!] 1 2 3 4 868
triple factorial: [n!!!, analogous to double factorial, A007661] 1 2 3 4 10 18 28 130 378 868 17290 1440188 756028 298995970 20741616378 57158060486
double dactorial: [(n!!)!!, analogous to double factorial, n!!, A006882] 1 2 3 384
dactorial: [“double factorial” portmanteau, (n!)!, NOT n!!, A006882] 1 2 720 2.228661662... x 105,436
octuple factorial: [n!!!!!!!!, analogous to double factorial, n!!, A006882] 1 2 3 4 5 6 7 8 9 20 33 48 65 84 105 128 153 560 1353 2688 4745 7728 11865 17408 24633 318080 1841433 7246848 22552985 59783808 140873145 303177728 606981753 101177431040 3390890224953 52516863909888 508637312834105 3574177251330
quadruple factorial : [n!!!!, analogous to double factorial, n!!, A006882] 1 2 3 4 5 12 21 32 45 192 525 1152 2205 37632 277725 1331712 4870845 1416317952 77132286525 1773462177792 23725150497405
quintuple factorial: [n!!!!!, analogous to double factorial, n!!, A006882] 1 2 3 4 5 6 14 24 36 50 66 266 696 1476 2750 4686 72086 487896 2185956 7576250 21982026 5196751826 238044946296 4778414563716 57399601943750 483209576974806
septuple factorial: [n!!!!!!!, analogous to double factorial, n!!, A006882] 1 2 3 4 5 6 7 8 18 30 44 60 78 98 120 450 1110 2244 4020 6630 10290 15240 205650 1239870 5051244 16188540 44003310 105956130 232364280 42293362050 1537286295990 25515101306244 262068940651380 1936291598979270
sextuple factorial: [n!!!!!!, analogous to double factorial, n!!1, A006882] 2 3 4 5 6 7 16 27 40 55 72 91 352 891 1840 3355 5616 8827 126016 799277 3396640 11276155 31573152 77968891 15880788352 638768592891 11537183669440 127151739240955 99684116654016 6079148431583230
tacforial: [ “New Ways to Get High”, Mpossibilities 70:3, Mar. 1999, ((((((nn-1)n-2)n-3)...)3)2)1 aka exponential factorial] 1 2 9 262144 6.206069874... x 10183,230 2.799137013... x 10507(1010)142,581
tractorial: [“triple factorial” portmanteau, ((n!)!)!] 1 2 2.228661662... x 105,436
repdigit: 1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 111 222 333 444 555 666 777 888 999 1111 2222 3333 4444 5555 6666 7777 8888 9999 11111 22222 33333 44444 55555 66666 77777 88888 99999 111111 222222 333333 444444 555555 666666
straight: [“Three Boxes”, Puzzle-M Apr. 1987, A028373] 1 4 7 11 14 17 41 44 47 71 74 77 111 114 117 141 144 147 171 174 177 411 414 417 441 444 447 471 474 477 711 714 717 741
L. sliced lucky pi: [Mpossibilities 64:2, pi without unlucky digits, 0, 2, 4, 5, 6, 8; sliced into ever increasing integers] 3 11 93 97 933 3337 9197 19399 37197 99371 9933117 7911337 9393173
LI. sliced odd pi: [Mpossibilities 64:2, pi without even digits; 81, Aug. 2002, sliced into ever increasing integers] 3 15 95 359 793 3333 7951 9719 9975 15979 59781 99353 117791 513379 395553 1753593 1919755 9317533 77315711
LII. sliced phhi: [Mpossibilities 64:3, pi:phi::phi:?, phhi = (?5 + 3)/(2ð); sliced into ever increasing integers] 8 33 34 61
LIII. sliced phhiplex: [Mpossibilities 64:3, pi:phi::phi:?; phhiplex = 10(?5 + 3)/(2ð), sliced into ever increasing integers] 6 8 13 120 984
LIV. sliced phiplex: [Mpossibilities 41:6, phiplex = 10(v5 + 1)/2, sliced into ever increasing integers] 41 49 86 519
LV. sliced piplex : [Mpossibilities 41:6, Dec. 1986, piplex = 10ð), sliced into ever increasing integers] 1385 4557 31367
LVI. sliced primeless pi : [Mpossibilities 64:3; 81, pi without 1, 3, 5, 7, sliced into ever increasing integers] 4 9 26 89 92 846 2648 2902 8849 69990 82094 944920 8640628 62089986 280482420 698248086 2820664098 4460908222 9408284840 28402098209 64462294894
90896442880 96694462846 482868622090 946486692460
4860442664829 602602494224800 66068848820920962
LVII. sliced Taliban pi: [Taliban pi sliced into ever increasing integers] 3 5 9 26 53 58 99 323 862 6338 32950 288969 3993505 820995 9230860 62862089 98628038 253206982 808653282 3066093860 9550582232 53590828850 280209385205 559662298959 3038962880956 6593362856823 38683652209095 68566923460386
LIX. sliced unlucky pi: [Mpossibilities 64:2, pi without lucky digits, 1, 3, 7, 9, sliced into ever increasing integers, Mpossibilities 81, Aug 2002, sliced into ever increasing integers] 4 5 26 55 82 84 626 4825 66482 60260 244224 5800660 6558848 8520206 28225405 64682506000 50548820466 52844654560 405206555502 868260585480 4462624565885
LX. sliced unrational: [“Newies” Mpossibilities 64, unrationals able to transform rationals to irrationals or vice versa, e. g., 3/2 - v2] 0 8 57 86 438
LXI. sliced valle pi: [Mpossibilities 66:6, valle, term from Panguingue, pi without 0, 2, 4, 6, 8, 9, sliced into ever increasing integers]
LXXI. untaliban: [with a, i, l, t] 2 3 5 6 8 9 22 23 25 26 28 29 32 33 35 36 38 39 52 53 55 56 58 59 62 63 65 66 68 69 82 83 85 86 88 89 92 93 95 96 98 99 222 223 225 226