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A117432 Let n be an integer consisting of m digits. Then n is a Phithy number if the n-th m-tuple in the decimal digits of golden ratio phi is string n.

%I A117432 #22 Feb 16 2025 08:33:00
%S A117432 1,20,63,104,7499430,9228401
%N A117432 Let n be an integer consisting of m digits. Then n is a Phithy number if the n-th m-tuple in the decimal digits of golden ratio phi is string n.
%H A117432 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">The Golden Ratio</a>
%e A117432 1 is a term because the first single digit in golden ratio phi is 1.
%e A117432 Number 20 is a term because the 20th pair of digits in phi is 20.
%e A117432 (cf. phi = 1.6180339887498948482045868343656381177203...)
%t A117432 PhithyNumbers[m_] := Module[{cc = m(10^m)+m, sol, aa}, sol = Partition[RealDigits[GoldenRatio, 10, cc] // First, m]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] Example: PhithyNumbers[3] produces all 3-digit Phithy numbers
%o A117432 (Python)
%o A117432 from sympy import S
%o A117432 def aupto(nn):
%o A117432   mm = len(str(nn))
%o A117432   phistr = str(S.GoldenRatio.n(nn*mm+1)).replace(".", "")[:-1]
%o A117432   for n in range(1, nn+1):
%o A117432     nstr = str(n)
%o A117432     m = len(nstr)
%o A117432     if phistr[(n-1)*m:n*m] == nstr: print(n, end=", ")
%o A117432 aupto(10**5) # _Michael S. Branicky_, Jan 20 2021
%Y A117432 Cf. A001622, A109513, A109514, A117431.
%K A117432 nonn,base,hard,more
%O A117432 0,2
%A A117432 _Colin Rose_, Mar 14 2006
%E A117432 a(4)-a(5) from _Michael S. Branicky_, Jan 21 2021