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 A065814 #34 Dec 30 2024 12:51:25
%S A065814 0,1,1,4,1,7,1,9,4,7,1,21,1,7,7,16,1,21,1,21,7,7,1,43,4,7,9,21,1,37,1,
%T A065814 25,7,7,7,56,1,7,7,43,1,37,1,21,21,7,1,73,4,21,7,21,1,43,7,43,7,7,1,
%U A065814 99,1,7,21,36,7,37,1,21,7,37,1,109,1,7,21,21,7,37,1,73,16,7,1,99,7,7,7,43
%N A065814 a(n) = tau(n)^2 - tau(n^2), where tau(n) = A000005(n).
%C A065814 If n = p^c = power of a prime, then a(n) = (c+1)^2 - (2c+1) = c^2. If n is squarefree with k prime factors then a(n) = 4^k - 3^k, so A065814(A002110(n)) = 4^n - 3^n = A005061(n). Terms depend on prime signature only.
%C A065814 If n is a prime (A000040), then a(n) = 1. If n is a semiprime (A001358), then a(n) = 4 + 3*ceiling(sqrt(n)) - 3*floor(sqrt(n)). If n is a triprime (A014612), then a(n) = 9 * floor(1/omega(n)) + 21 * (1 - (omega(n) mod 2)) + 37 * floor(omega(n)/3), n > 1. - _Wesley Ivan Hurt_, May 24 2013
%H A065814 Harry J. Smith, <a href="/A065814/b065814.txt">Table of n, a(n) for n = 1..1000</a>
%F A065814 a(n) = A000005(n)^2 - A000005(n^2).
%F A065814 G.f.: Sum_{n>=1} A000005(n^2)*x^(2*n)/(1-x^n). - _Mircea Merca_, Feb 26 2014
%F A065814 a(n) = A035116(n) - A048691(n). - _Amiram Eldar_, Apr 25 2024
%t A065814 a[n_] := DivisorSigma[0, n]^2 - DivisorSigma[0, n^2]; Array[a, 100] (* _Amiram Eldar_, Apr 25 2024 *)
%o A065814 (PARI) a(n) = { numdiv(n)^2 - numdiv(n^2) } \\ _Harry J. Smith_, Oct 31 2009
%Y A065814 Cf. A000005, A000040, A000290, A000961, A001358, A002110, A005061, A014612, A035116, A048691.
%K A065814 nonn
%O A065814 1,4
%A A065814 _Labos Elemer_, Nov 22 2001