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.
%I A092591 #33 Jun 18 2025 00:50:20
%S A092591 0,1,2,3,4,6,7,12,15,16,18,30,31,60,88,106,126,520,606,1278,2202,2280,
%T A092591 3216,4252,4422,9688,9940,11212,19936,21700,23208,44496,86242,110502,
%U A092591 132048,216090,756838,859432,1257786,1398268,2976220,3021376,6972592,13466916,20996010,24036582,25964950,30402456,32582656,37156666,42643800,43112608,57885160
%N A092591 Exponents m such that 1-A065395(2^m) is a power of 2, where A065395(n) = sigma(phi(n)) - phi(sigma(n)).
%C A092591 A000043(k) - 1 is a term for all k >= 1. - _Amiram Eldar_, Aug 22 2019
%C A092591 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
%F A092591 If there are only 5 Fermat primes (A019434), then for n >= 14, a(n) = A000043(n-5) - 1. - _Max Alekseyev_, Jun 14 2025
%e A092591 At exponents m=1, 3, 7, 15, 31: 1-A065395(2^m)=2.
%e A092591 While at m=2, 4, 6, 12, 16, 18, 30, 60, 88, 106, 126: 1-A065395(2^m)=2^m.
%t A092591 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 *)
%o A092591 (PARI) f(n) = sigma(eulerphi(n)) - eulerphi(sigma(n)); \\ A065395
%o A092591 ispp2(k) = k == 2^valuation(k,2);
%o A092591 isok(n) = ispp2(1-f(2^n)); \\ _Michel Marcus_, Aug 22 2019, Jun 16 2025
%Y A092591 Cf. A000010, A000203, A000043, A065395, A092584, A092585, A092586, A092587, A092588, A092589, A092590.
%K A092591 nonn
%O A092591 1,3
%A A092591 _Labos Elemer_, Mar 03 2004
%E A092591 Name and example edited by _Michel Marcus_, Aug 22 2019
%E A092591 a(18)-a(19) from _Amiram Eldar_, Aug 23 2019
%E A092591 a(1)=0 inserted and terms a(20) onward added by _Max Alekseyev_, Jun 14 2025