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 A104905 #12 Feb 24 2024 01:10:41
%S A104905 1,3,14,42
%N A104905 Numbers m such that d(m)*phi(m) = sigma(m), where d(m) is number of positive divisors of m.
%C A104905 d(m)*phi(m) is the product of f(p^k) = (p^k - p^(k-1))*(1+k), while sigma(m) is the product of g(p^k) = (p^(k+1)-1)/(p-1) taken over all prime powers p^k in the factorization of m. We have f(p^k) < g(p^k) for p=2 and k=1 or 2; f(p^k) = g(p^k) for p=3, k=1; and f(p^k) > g(p^k) in all other cases. Furthermore, f(2)/g(2) = 2/3 and f(2^2)/g(2^2) = 6/7, while f(p^k)/g(p^k) > f(p)/g(p) and for p > 7, f(p)/g(p) > 3/2. It easily follows that 1,3,14,42 are the only terms of this sequence. - _Max Alekseyev_, Feb 08 2010
%e A104905 42 is in the sequence because d(42)=8; phi(42)=12; sigma(42)=96 & 8*12=96.
%t A104905 Do[If[DivisorSigma[0, n]*EulerPhi[n] == DivisorSigma[1, n], Print[n]], {n, 530000000}]
%Y A104905 Cf. A063903, A104904, A104906.
%K A104905 nonn,full,fini
%O A104905 1,2
%A A104905 _Farideh Firoozbakht_, Apr 13 2005
%E A104905 Keywords full, fini from _Max Alekseyev_, Feb 08 2010