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

0,2

Links

Table of n, a(n) for n=0..5.

Eric Weisstein's World of Mathematics, The Golden Ratio

Example

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...)

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

Prog

(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

Crossrefs

Cf. A001622, A109513, A109514, A117431.

Sequence in context: A117431 A159504 A182468 * A033577 A262486 A187156

Adjacent sequences: A117429 A117430 A117431 * A117433 A117434 A117435

Keyword

nonn,base,hard,more

Author

Colin Rose, Mar 14 2006

Extensions

a(4)-a(5) from Michael S. Branicky, Jan 21 2021

Status

approved