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A290977
First n-digit number to appear twice in a row in the decimal expansion of Pi.
2
3, 59, 209, 9314, 64015, 886287, 7348278, 85105027
OFFSET
1,1
COMMENTS
209209 and 305305 appear in Pi before any 2-digit number appears twice in a row.
a(n) (n >= 1) begins at the following decimal places: 24, 413, 326, 8239, 107472, 1632152, 9719518. - Robert G. Wilson v, Aug 23 2017
EXAMPLE
a(1) = 3 because 3 is the first 1-digit number to appear twice in a row in the decimal expansion of Pi = 3.14159265358979323846264(33)...
MATHEMATICA
With[{s = Rest@ First@ RealDigits[N[Pi, 10^4]]}, Keys@ Merge[#, Identity] &@ Table[If[Length@ # > 0, TakeSmallest[#, 1], 0 -> 0] &@ Sort[Map[#[[1, 1]] &, DeleteCases[#, {}]]] &@ Map[SequenceCases[#, {a_, b_} /; b == a + n] &, KeyMap[FromDigits, PositionIndex@ Partition[s, n, 1]]], {n, 4}]] (* Michael De Vlieger, Aug 16 2017 *)
pi = StringDrop[ ToString[ N[Pi, 1632200]], 2]; f[n_] := Block[{k = 1}, While[ StringTake[pi, {k, k +n -1}] != StringTake[pi, {k +n, k +2n -1}], k++]; k]; Array[f, 6] (* Robert G. Wilson v, Aug 17 2017 *)
PROG
(PARI) eva(n) = subst(Pol(n), x, 10)
pistring(n) = default(realprecision, n+10); my(x=Pi); floor(x*10^n)
pidigit(n) = pistring(n)-10*pistring(n-1)
consecpidigits(pos, len) = my(v=vector(len)); for(k=1, len, v[k]=pidigit(pos+k)); v
a(n) = my(v=[], w=[], x=1); while(1, v=consecpidigits(x, n); w=consecpidigits(x+n, n); if(v==w, return(eva(v))); x++) \\ Felix Fröhlich, Aug 16 2017
(Python)
from sympy import S
# download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then
# with open('pi-billion.txt', 'r') as f: pi_digits = f.readline()
pi_digits = str(S.Pi.n(3*10**5+2))[:-2] # alternative to above
pi_digits = pi_digits.replace(".", "")
def a(n):
for k in range(1, len(pi_digits)-n):
s = pi_digits[k:k+2*n]
if s[0] != 0 and s[:len(s)//2] == s[len(s)//2:]:
return int(s[:len(s)//2])
print([a(n) for n in range(1, 6)]) # Michael S. Branicky, Jan 10 2022
KEYWORD
base,more,nonn
AUTHOR
Bobby Jacobs, Aug 16 2017
EXTENSIONS
a(6) from Robert G. Wilson v, Aug 19 2017
a(7) from Bobby Jacobs, Aug 22 2017
a(8) from Michael S. Branicky, Jan 10 2022
STATUS
approved