A068987 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).

2, 149, 1925, 13808, 49703, 2458886, 9470345, 186557267, 523551503, 191278379840, 4368196101672

Offset: 1

Author

Joseph L. Pe, Apr 01 2002

Keywords

• nonn
• base
• more

Comments

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

This sequence uses position = 1 for the initial digit 3 of Pi, while A121280(n) = a(n)-1 starts counting at 0, as does the "Pi search page" and sequences A035117, A050279 - A050287, A048940, A096755 - A096763. - M. F. Hasler, Mar 18 2017

a(10) > 2*10^9. - M. F. Hasler, Apr 13 2019

a(12) > 22*10^12. - Dmitry Petukhov, Jan 29 2020

References

• Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, p. 32.

Links

• D. G. Andersen, The Pi-Search Page
• SubIdiom.com, Irrational numbers search engine: π = 3.14159.... (Search within 2*10^9 digits, since at least 2009, maybe 2002.)
• Peter Trüb, 22.4 trillion digits of pi

Crossrefs

First occurrence of n times the same digit: A035117 (n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).

First occurrence of exactly n times the same digit: A096755 (exactly n '1's), A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763 (exactly n '9's), A050279 (exactly n '0's).

First occurrence of n: A176341; of concatenate(1,...,n): A121280 = A068987 - 1.

Cf. A000796: Decimal expansion (or digits) of Pi.

Programs

Mathematica

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}]

Formula

a(n) = A121280(n) + 1. - M. F. Hasler, Apr 13 2019

Extensions

More terms from Rick L. Shepherd, Oct 10 2002

a(8) from Rick L. Shepherd, Aug 06 2006

Additional term a(9), using subidiom search engine, from M. F. Hasler, Apr 13 2019

a(10)-a(11) from Dmitry Petukhov, Jan 16 2020