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.
1, 20, 63, 104, 7499430, 9228401
Offset: 0
Examples
1 is a term because the first single digit in golden ratio phi is 1. Number 20 is a term because the 20th pair of digits in phi is 20. (cf. phi = 1.6180339887498948482045868343656381177203...)
Links
- Eric Weisstein's World of Mathematics, The Golden Ratio
Programs
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Mathematica
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 -
Python
from sympy import S def aupto(nn): mm = len(str(nn)) phistr = str(S.GoldenRatio.n(nn*mm+1)).replace(".", "")[:-1] for n in range(1, nn+1): nstr = str(n) m = len(nstr) if phistr[(n-1)*m:n*m] == nstr: print(n, end=", ") aupto(10**5) # Michael S. Branicky, Jan 20 2021
Extensions
a(4)-a(5) from Michael S. Branicky, Jan 21 2021