cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A092591 Exponents m such that 1-A065395(2^m) is a power of 2, where A065395(n) = sigma(phi(n)) - phi(sigma(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 30, 31, 60, 88, 106, 126, 520, 606, 1278, 2202, 2280, 3216, 4252, 4422, 9688, 9940, 11212, 19936, 21700, 23208, 44496, 86242, 110502, 132048, 216090, 756838, 859432, 1257786, 1398268, 2976220, 3021376, 6972592, 13466916, 20996010, 24036582, 25964950, 30402456, 32582656, 37156666, 42643800, 43112608, 57885160
Offset: 1

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Author

Labos Elemer, Mar 03 2004

Keywords

Comments

A000043(k) - 1 is a term for all k >= 1. - Amiram Eldar, Aug 22 2019
Let 2^k = 1-A065395(2^m) = phi(2^(m+1)-1) - 2^m + 2. If k = 0, then phi(2^(m+1)-1) is odd, implying extraneous m = 0. If k = 1, then phi(2^(m+1)-1) = 2^m, meaning that 2^(m+1)-1 is a product of distinct Fermat primes (A019434), which also a term of A050474. The five known Fermat primes give m in {0, 1, 3, 7, 15, 31}. If k >= 2, then phi(2^(m+1)-1) == 2 (mod 4), implying that 2^(m+1)-1 is a prime power, and by Mihăilescu's theorem, 2^(m+1)-1 must be just a prime, that is, m+1 is a term of A000043 and k = m. Hence, unless there exist other Fermat primes, this sequence is the union of {0, 1, 3, 7, 15, 31} and terms of A000043 decreased by 1. - Max Alekseyev, Jun 14 2025

Examples

			At exponents m=1, 3, 7, 15, 31: 1-A065395(2^m)=2.
While at m=2, 4, 6, 12, 16, 18, 30, 60, 88, 106, 126: 1-A065395(2^m)=2^m.

Crossrefs

Cf. A000010, A000203, A000043, A065395, A092584, A092585, A092586, A092587, A092588, A092589, A092590.

Programs

  • Mathematica
    f[n_] := DivisorSigma[1, EulerPhi[n]] - EulerPhi[DivisorSigma[1, n]]; pow2Q[n_] := n == 2^IntegerExponent[n, 2]; aQ[n_] := pow2Q[1 - f[2^n]]; Select[Range[0, 130], aQ] (* Amiram Eldar, Aug 22 2019 *)
  • PARI
    f(n) = sigma(eulerphi(n)) - eulerphi(sigma(n)); \\ A065395
ispp2(k) = k == 2^valuation(k,2);
isok(n) = ispp2(1-f(2^n)); \\ Michel Marcus, Aug 22 2019, Jun 16 2025

Formula

If there are only 5 Fermat primes (A019434), then for n >= 14, a(n) = A000043(n-5) - 1. - Max Alekseyev, Jun 14 2025

Extensions

Name and example edited by Michel Marcus, Aug 22 2019
a(18)-a(19) from Amiram Eldar, Aug 23 2019
a(1)=0 inserted and terms a(20) onward added by Max Alekseyev, Jun 14 2025
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