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

%I A197123 #59 Dec 29 2024 13:59:12
%S A197123 1,26,592,582,60943,949129,8530614,52637962,201890888,4392366484,
%T A197123 89879780761,756130190263,3186120489507,18220874234996,
%U A197123 276854551127715,8230687217052243,93415455347042966,13724950651727463,1350168131352524443,84756845106452435773,585270898631522188621,2761994111668451704865,64722721994615606186022,307680366924568801265656
%N A197123 a(n) is the first n-digit substring to repeat in the decimal expansion of Pi.
%C A197123 a(4) is written in the sequence as a 3-digit number 582 because the repeating substring is the 4-digit number 0582.
%C A197123 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
%H A197123 Dave Andersen, <a href="http://www.angio.net/pi/piquery">The Pi-Search Page.</a>
%H A197123 David H. Bailey, <a href="http://crd.lbl.gov/~dhbailey/dhbpapers/pi.pdf">The computation of pi to 29,360,000 decimal digits using Borweins' quartically convergent algorithm</a>, Mathematics of Computation 50 (1988), pp. 283-296.
%H A197123 MIT Student Information Processing Board, <a href="https://stuff.mit.edu/afs/sipb/contrib/pi/">One billion digits of Pi</a>.
%H A197123 Jonas Schmitz, <a href="https://jonas.sh/world-record-finding-longest-reapting-sequence-in-decimal-expanson-of-pi-2/">World Record: Finding the first repeating 24-digit substring of Pi</a>.
%H A197123 Jeff Sponaugle, <a href="https://sponaugle.com/wp/math_pi_repeat/">Calculating a(19)-a(22) in A197123</a>.
%e A197123 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.
%e A197123 From _Bobby Jacobs_, Dec 24 2016: (Start)
%e A197123 1 appears at positions 1 and 3.
%e A197123 26 appears at positions 6 and 21.
%e A197123 592 appears at positions 4 and 61.
%e A197123 0582 appears at positions 50 and 132.
%e A197123 60943 appears at positions 397 and 551.
%e A197123 949129 appears at positions 496 and 1296.
%e A197123 8530614 appears at positions 4167 and 4601.
%e A197123 ... (End)
%o A197123 (Python)
%o A197123 # download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then
%o A197123 # with open('pi-billion.txt', 'r') as f: digits_of_pi = f.readline()
%o A197123 from sympy import S; digits_of_pi = str(S.Pi.n(3*10**5)) # alternatively
%o A197123 def a(n):
%o A197123   global digits_of_pi
%o A197123   seen = set()
%o A197123   for i in range(2, len(digits_of_pi)-n):
%o A197123     ss = digits_of_pi[i:i+n]
%o A197123     if ss in seen: return int(ss)
%o A197123     seen.add(ss)
%o A197123 for n in range(1, 11):
%o A197123   print(a(n), end=", ") # _Michael S. Branicky_, Jan 26 2021
%Y A197123 Cf. A000796 (Pi), A159345 (the number of digits of Pi required to include the repeated string), A279860.
%K A197123 base,nonn
%O A197123 1,2
%A A197123 _Peter de Rivaz_, Oct 10 2011
%E A197123 a(16)-a(18) from _Clive Tooth_, Mar 06 2012
%E A197123 a(19)-a(22) from _Jeff Sponaugle_, Aug 22 2024
%E A197123 a(23) from _Jeff Sponaugle_, Sep 23 2024
%E A197123 a(24) from _Jonas Schmitz_,  Dec 16 2024