Search: a121280 -id:a121280
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A035117
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a(n) is the starting position of the first occurrence of a string of at least n 1's in the decimal expansion of Pi.
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+0
21
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1, 94, 153, 12700, 32788, 255945, 4657555, 159090113, 812432526, 3961184001, 15647738228, 1041032609981, 3907688331257, 68635742334547
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OFFSET
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1,2
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COMMENTS
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Presently identical to A096755, which is the first occurrences of exactly n 1's in the digits of Pi. Will differ as soon as there's some a(n) = a(n+1) and equivalently, A035117(n) > A035117(n+1). - M. F. Hasler, Mar 17 2017
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LINKS
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CROSSREFS
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Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A176341 (first occurrence of n in Pi's digits).
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KEYWORD
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nonn,base,more
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AUTHOR
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Leonardo Bitran (lbitran(AT)reuna.cl)
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EXTENSIONS
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More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
a(12) from Yasumasa Kanada Laboratory, 2002 and a(13) from Shigeru Kondo, 2011, added by Dmitry Petukhov, Dec 27 2019
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STATUS
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approved
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A037001
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Positions of the digit '2' in the decimal expansion of Pi (where positions 0, 1, 2,... refer to the digits 3, 1, 4,...).
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+0
26
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6, 16, 21, 28, 33, 53, 63, 73, 76, 83, 89, 93, 102, 112, 114, 135, 136, 140, 149, 160, 165, 173, 185, 186, 203, 221, 229, 241, 244, 260, 275, 280, 289, 292, 298, 302, 326, 329, 333, 335, 337, 354, 374, 380, 406, 423, 435, 456, 462, 477, 479, 484, 485, 500
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OFFSET
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1,1
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COMMENTS
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The first few primes in this sequence are 53, 73, 83, 89, 149, 173, 229, 241, 337, 479, 571, 613, 661, 757, 829, 877, 911, 977, 991, ... - M. F. Hasler, Jul 28 2024
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LINKS
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FORMULA
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a(n) ~ 10*n if Pi is normal, as generally assumed. - M. F. Hasler, Jul 28 2024
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MATHEMATICA
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Flatten @ Position[ RealDigits[Pi - 3, 10, 500][[1]], 2] (* Robert G. Wilson v, Mar 07 2011 *)
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PROG
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(PARI) A037001_upto(N=999, d=2)={localprec(N+20); [i-1|i<-[1..#N=digits(Pi\10^-N)], N[i]==d]} \\ M. F. Hasler, Jul 28 2024
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CROSSREFS
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Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A053746 (= a(n) + 1: the same with different offset).
Cf. A037000, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A037008 (similar for digits 1, ..., 9 and 0).
Cf. A176341: first occurrence of n in Pi's digits.
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KEYWORD
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nonn,base
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AUTHOR
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Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br)
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STATUS
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approved
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A048940
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Starting position of the first occurrence of a string of at least n '9's in the decimal expansion of Pi.
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+0
20
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5, 44, 762, 762, 762, 762, 1722776, 36356642, 564665206, 20148132310, 27014073304, 897831316556, 5758910552709, 5758910552709
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OFFSET
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1,1
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COMMENTS
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Pi digits 3,1,4,... are indexed 0,1,2,... Note that this is different from other sequences such as A049522, A084073 which use indices 1,2,3,... For example, the position of the curious accumulation of six 9s is called 762 here; the same position is called 763 in A049522, A084073. - Jeppe Stig Nielsen, Aug 21 2017
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LINKS
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David G. Andersen, The Pi-Search Page. (Yields, as of today, an incorrect result of 66780105 for the first occurrence of eight "9"s. - M. F. Hasler, Mar 19 2017)
Eric Weisstein's World of Mathematics, Pi Digits
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MATHEMATICA
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Module[{m, nn = 7}, m = First@ RealDigits@ N[Pi, 10^nn]; Array[ SequencePosition[m, ConstantArray[9, #]][[1, 1]] - 1 &, nn]] (* Michael De Vlieger, Mar 20 2017 *)
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CROSSREFS
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Cf. A000796: Decimal expansion (or digits) of Pi.
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KEYWORD
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nonn,base,more
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AUTHOR
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EXTENSIONS
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More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
a(12) from Yasumasa Kanada Laboratory, 2002 and a(13)-a(14) from Shigeru Kondo, 2011 added by Dmitry Petukhov, Dec 23 2019
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STATUS
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approved
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A050279
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a(n) is the starting position of the first occurrence of a string of at least n '0's in the decimal expansion of Pi.
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+0
21
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32, 307, 601, 13390, 17534, 1699927, 3794572, 172330850, 2542542102, 8324296435, 371247087572, 1755524129973, 3186699229890, 6381820482331
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OFFSET
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1,1
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COMMENTS
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At least up to a(10), also the starting position of the first occurrence of a string of exactly n '0's in the decimal expansion of Pi, cf. A096764. - M. F. Hasler, Mar 19 2017, edited Sep 03 2017
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REFERENCES
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Shigeru Kondo, calculation of Pi to 12.8 * 10^9 digits, using the program PiFast of Xavier Gourdon
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LINKS
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Eric Weisstein's World of Mathematics, Pi Digits
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CROSSREFS
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Cf. A000796: Decimal expansion (or digits) of Pi.
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KEYWORD
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nonn,base,more
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AUTHOR
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EXTENSIONS
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More terms from Colin B. Martin (martinc(AT)ram.net.au), Nov 25 2001
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STATUS
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approved
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A050286
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Starting position of the first occurrence of a string of at least n '7's in the decimal expansion of Pi.
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+0
13
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13, 559, 1589, 1589, 162248, 399579, 3346228, 24658601, 24658601, 22869046249, 165431035708, 368299898266, 10541103245815, 14793486898235, 46970519777308
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OFFSET
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1,1
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COMMENTS
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a(10) > 2*10^9 according to the SubIdiom.com/pi search engine. - M. F. Hasler, Apr 13 2019
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LINKS
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Eric Weisstein's World of Mathematics, Pi Digits.
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FORMULA
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CROSSREFS
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Cf. A000796: Decimal expansion (or digits) of Pi.
Cf. A176341 (first occurrence of n).
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KEYWORD
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nonn,base,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A050287
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Starting position of the first occurrence of a string of at least n '8's in the decimal expansion of Pi.
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+0
13
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11, 34, 4751, 4751, 213245, 222299, 4722613, 46663520, 46663520, 3040319543, 159999448572, 1141385905180, 2164164669332, 91250566353705
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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LINKS
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CROSSREFS
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Cf. A000796: Decimal expansion (or digits) of Pi.
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KEYWORD
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nonn,base,more
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AUTHOR
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EXTENSIONS
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More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
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STATUS
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approved
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A053746
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Positions of '2's in the decimal expansion of Pi, where positions 1, 2, 3, ... correspond to digits 3, 1, 4, ...
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+0
7
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7, 17, 22, 29, 34, 54, 64, 74, 77, 84, 90, 94, 103, 113, 115, 136, 137, 141, 150, 161, 166, 174, 186, 187, 204, 222, 230, 242, 245, 261, 276, 281, 290, 293, 299, 303, 327, 330, 334, 336, 338, 355, 375, 381, 407
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graph;
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listen;
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OFFSET
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1,1
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COMMENTS
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See A037001 for the variant where digits 3, 1, 4, ... correspond to positions 0, 1, 2, ... - M. F. Hasler, Jul 28 2024
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LINKS
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FORMULA
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EXAMPLE
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Pi = 3.1415926... where the first '2' occurs as the 7th digit.
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MATHEMATICA
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Flatten[Position[RealDigits[Pi, 10, 1000][[1]], 2]] (* Vincenzo Librandi, Oct 07 2013 *)
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PROG
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(PARI) A053746_upto(N=999)={localprec(N+20); select(d->d==2, digits(Pi\10^-N), 1)} \\ M. F. Hasler, Jul 28 2024
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CROSSREFS
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Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A037001 (= a(n) - 1: the same with different offset).
Cf. A176341: first occurrence of n in Pi's digits.
Cf. A088566 (primes in this sequence).
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A068987
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a(n) is the first position in the digit sequence 3,1,4,1,5,9,.... of Pi where the pattern "1,2,...,n" occurs (where position of the initial 3 is 1).
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15
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2, 149, 1925, 13808, 49703, 2458886, 9470345, 186557267, 523551503, 191278379840, 4368196101672
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OFFSET
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1,1
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COMMENTS
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1. We may never know if a(n) is defined for all n.
2. We split up the digits of any number > 9 in the pattern, e.g., if n = 11, we search for the pattern "1,2,3,4,5,6,7,8,9,1,0,1,1".
3. The pattern "1,2,3,4,5,6" does not occur before the 100,000th term in the digit sequence of Pi.
Two more terms a(6) and a(7) were found via the referenced Pi-Search link [Andersen], through which 100 million digits of Pi are currently available. - Rick L. Shepherd, Oct 10 2002
200 million digits now available at Pi-Search page. - Rick L. Shepherd, Aug 06 2006
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REFERENCES
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Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, p. 32.
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LINKS
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FORMULA
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MATHEMATICA
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p = ToString[N[Pi, 50000]/10]; t = {1, 12, 123, 1234, 12345}; g[n_] := StringPosition[p, ToString[n]][[1]][[1]] - 2; Table[g[t[[i]]], {i, 1, 5}]
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CROSSREFS
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Cf. A000796: Decimal expansion (or digits) of Pi.
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KEYWORD
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nonn,base,more
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AUTHOR
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EXTENSIONS
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Additional term a(9), using subidiom search engine, from M. F. Hasler, Apr 13 2019
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STATUS
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approved
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A096761
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Position of first occurrence of exactly n consecutive sevens in a row in the decimal expansion of Pi.
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+0
21
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13, 559, 4575, 1589, 162248, 399579, 3346228, 82144203, 24658601, 22869046249, 165431035708, 368299898266, 10541103245815, 14793486898235, 46970519777308
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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LINKS
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CROSSREFS
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Cf. A000796 (decimal expansion (or digits) of Pi).
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KEYWORD
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nonn,base,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A096762
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Position of first occurrence of exactly n consecutive '8's in a row in the decimal expansion of Pi.
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+0
21
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11, 34, 4985, 4751, 213245, 222299, 4722613, 239798471, 46663520, 3040319543, 159999448572, 1141385905180, 2164164669332, 91250566353705
(list;
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listen;
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OFFSET
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1,1
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COMMENTS
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a(8) > 2*10^8, a(9) = 46663520, a(10) = 3040319543.
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LINKS
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CROSSREFS
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Cf. A000796: Decimal expansion (or digits) of Pi.
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KEYWORD
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base,nonn,more
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AUTHOR
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EXTENSIONS
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a(8) via SubIdiom.com/pi search engine from M. F. Hasler, Apr 13 2019
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STATUS
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approved
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