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A117431
String n is at position n in decimal digits of golden ratio (phi).
1
1, 20, 62, 9956
OFFSET
1,2
COMMENTS
The next such number is greater than 10^7. Not only does number 20 occur at the 20th digit, but it occurs again as the 20th pair of digits (cf. A117432).
LINKS
Eric Weisstein's World of Mathematics, The Golden Ratio
EXAMPLE
1 is a term because the first digit in the golden ratio phi is 1. (phi = 1.6180339887498948482045 ...)
MATHEMATICA
StringFinder[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[(1+Sqrt[5])/2, 10, cc] // First, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i, Length[sol]}]; ] (* Example: StringFinder[2] produces all 2-digit members of the sequence. *)
PROG
(Python)
from sympy import S
def aupto(nn):
phistr = str(S.GoldenRatio.n(nn+len(str(nn))+1)).replace(".", "")[:-1]
for n in range(1, nn+1):
nstr = str(n)
if phistr[n-1:n-1+len(nstr)] == nstr: print(n, end=", ")
aupto(10**5) # Michael S. Branicky, Jan 20 2021
CROSSREFS
KEYWORD
nonn,base,hard,more
AUTHOR
Colin Rose, Mar 14 2006
STATUS
approved