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

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

Author

Labos Elemer, Mar 03 2004

Keywords

• nonn

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