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 A343443 #26 Feb 21 2025 17:48:58
%S A343443 1,3,3,4,3,9,3,5,4,9,3,12,3,9,9,6,3,12,3,12,9,9,3,15,4,9,5,12,3,27,3,
%T A343443 7,9,9,9,16,3,9,9,15,3,27,3,12,12,9,3,18,4,12,9,12,3,15,9,15,9,9,3,36,
%U A343443 3,9,12,8,9,27,3,12,9,27,3,20,3,9,12,12,9,27,3,18
%N A343443 If n = Product (p_j^k_j) then a(n) = Product (k_j + 2), with a(1) = 1.
%C A343443 Inverse Moebius transform of A056671.
%C A343443 a(n) depends only on the prime signature of n (see formulas). - _Bernard Schott_, May 03 2021
%H A343443 Amiram Eldar, <a href="/A343443/b343443.txt">Table of n, a(n) for n = 1..10000</a>
%H A343443 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A343443 a(n) = 2^omega(n) * tau_3(n) / tau(n), where omega = A001221, tau = A000005 and tau_3 = A007425.
%F A343443 a(n) = Sum_{d|n, gcd(d, n/d) = 1} tau(d).
%F A343443 From _Bernard Schott_, May 03 2021: (Start)
%F A343443 a(p^k) = k+2 for p prime, or signature [k].
%F A343443 a(A006881(n)) = 9 for signature [1, 1].
%F A343443 a(A054753(n)) = 12 for signature [2, 1].
%F A343443 a(A065036(n)) = 15 for signature [3, 1].
%F A343443 a(A085986(n)) = 16 for signature [2, 2].
%F A343443 a(A178739(n)) = 18 for signature [4, 1].
%F A343443 a(A143610(n)) = 20 for signature [3, 2].
%F A343443 a(A007304(n)) = 27 for signature [1, 1, 1]. (End)
%F A343443 Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)). - _Vaclav Kotesovec_, Feb 11 2023
%F A343443 From _Amiram Eldar_, Sep 01 2023: (Start)
%F A343443 a(n) = A000005(A064549(n)).
%F A343443 a(n) = A363194(A348018(n)). (End)
%t A343443 a[1] = 1; a[n_] := Times @@ ((#[[2]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 80}]
%t A343443 a[n_] := Sum[If[GCD[d, n/d] == 1, DivisorSigma[0, d], 0], {d, Divisors[n]}]; Table[a[n], {n, 80}]
%o A343443 (PARI) a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, numdiv(d))) \\ _Andrew Howroyd_, Apr 15 2021
%o A343443 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Feb 11 2023
%o A343443 (Python)
%o A343443 from math import prod
%o A343443 from sympy import factorint
%o A343443 def A343443(n): return prod(e+2 for e in factorint(n).values()) # _Chai Wah Wu_, Feb 21 2025
%Y A343443 Cf. A000005, A001221, A005361, A007425, A025847, A034444, A056671, A064549, A074816, A107758, A107759, A348018, A363194.
%K A343443 nonn,easy,mult
%O A343443 1,2
%A A343443 _Ilya Gutkovskiy_, Apr 15 2021