A197123 a(n) is the first n-digit substring to repeat in the decimal expansion of Pi.

1, 26, 592, 582, 60943, 949129, 8530614, 52637962, 201890888, 4392366484, 89879780761, 756130190263, 3186120489507, 18220874234996, 276854551127715, 8230687217052243, 93415455347042966, 13724950651727463, 1350168131352524443, 84756845106452435773, 585270898631522188621, 2761994111668451704865, 64722721994615606186022, 307680366924568801265656

Offset: 1

Author

Peter de Rivaz, Oct 10 2011

Keywords

• base
• nonn

Comments

a(4) is written in the sequence as a 3-digit number 582 because the repeating substring is the 4-digit number 0582.

a(18) should also have a leading zero: 013724950651727463. This value starts at digit 378,355,223 and at digit 1,982,424,643. This computation was performed by Richard Tobin. - Clive Tooth, Mar 06 2012

Examples

			For n=2 the a(2)=26 solution is because if we look at all the 2-digit substrings 14,41,15,59,92,26,... of the decimal expansion of Pi=3.1415926535897932384626 we find that the first 2-digit substring to appear twice is 26.
From _Bobby Jacobs_, Dec 24 2016: (Start)
1 appears at positions 1 and 3.
26 appears at positions 6 and 21.
592 appears at positions 4 and 61.
0582 appears at positions 50 and 132.
60943 appears at positions 397 and 551.
949129 appears at positions 496 and 1296.
8530614 appears at positions 4167 and 4601.
... (End)
		

Links

• Dave Andersen, The Pi-Search Page.
• David H. Bailey, The computation of pi to 29,360,000 decimal digits using Borweins' quartically convergent algorithm, Mathematics of Computation 50 (1988), pp. 283-296.
• MIT Student Information Processing Board, One billion digits of Pi.
• Jonas Schmitz, World Record: Finding the first repeating 24-digit substring of Pi.
• Jeff Sponaugle, Calculating a(19)-a(22) in A197123.

Crossrefs

Cf. A000796 (Pi), A159345 (the number of digits of Pi required to include the repeated string), A279860.

Programs

Python

# download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then
# with open('pi-billion.txt', 'r') as f: digits_of_pi = f.readline()
from sympy import S; digits_of_pi = str(S.Pi.n(3*10**5)) # alternatively
def a(n):
  global digits_of_pi
  seen = set()
  for i in range(2, len(digits_of_pi)-n):
    ss = digits_of_pi[i:i+n]
    if ss in seen: return int(ss)
    seen.add(ss)
for n in range(1, 11):
  print(a(n), end=", ") # Michael S. Branicky, Jan 26 2021

Extensions

a(16)-a(18) from Clive Tooth, Mar 06 2012

a(19)-a(22) from Jeff Sponaugle, Aug 22 2024

a(23) from Jeff Sponaugle, Sep 23 2024

a(24) from Jonas Schmitz, Dec 16 2024