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A306379 Dirichlet convolution of psi(n) with itself.

%I A306379 #34 Feb 16 2025 08:33:55
%S A306379 1,6,8,21,12,48,16,60,40,72,24,168,28,96,96,156,36,240,40,252,128,144,
%T A306379 48,480,96,168,168,336,60,576,64,384,192,216,192,840,76,240,224,720,
%U A306379 84,768,88,504,480,288,96,1248,176,576,288,588,108,1008,288,960,320
%N A306379 Dirichlet convolution of psi(n) with itself.
%C A306379 For n>1, a(n)>=2*n+2 with equality iff n is prime. - _Robert Israel_, Feb 28 2019
%C A306379 Sum_{k>=1} 1/a(k) diverges. - _Vaclav Kotesovec_, Sep 20 2020
%H A306379 Robert Israel, <a href="/A306379/b306379.txt">Table of n, a(n) for n = 1..10000</a>
%H A306379 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DedekindFunction.html">Dedekind Function</a>.
%H A306379 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dedekind_psi_function">Dedekind psi function</a>.
%F A306379 a(n) = Sum_{d|n} psi(d) * psi(n/d).
%F A306379 From _Jianing Song_, Apr 28 2019: (Start)
%F A306379 Multiplicative with a(p^e) = (e-1)*(p+1)^2*p^(e-2) + 2*(p+1)*p^(e-1).
%F A306379 Dirichlet g.f.: (zeta(s) * zeta(s-1) / zeta(2*s))^2. (End)
%F A306379 Sum_{k=1..n} a(k) ~ 225*(2*log(n) + 4*gamma - 1 + 24*zeta'(2)/Pi^2 - 720*zeta'(4)/Pi^4) * n^2 / (4*Pi^4), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Sep 20 2020
%p A306379 psi:= proc(n) local p; option remember;  n*mul(1+1/p, p = numtheory:-factorset(n)): end proc:
%p A306379 f:= proc(n) local d;
%p A306379    add(psi(d)*psi(n/d),d = numtheory:-divisors(n))
%p A306379 end proc:
%p A306379 map(f, [$1..100]); # _Robert Israel_, Feb 28 2019
%t A306379 psi[n_] := n Times @@ (1+1/FactorInteger[n][[All, 1]]); psi[1] = 1;
%t A306379 a[n_] := Sum[psi[d] psi[n/d], {d, Divisors[n]}];
%t A306379 Array[a, 100] (* _Jean-François Alcover_, Oct 16 2020 *)
%t A306379 f[p_, e_] := (e-1)*(p+1)^2*p^(e-2) + 2*(p+1)*p^(e-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 22 2020 *)
%o A306379 (PARI) f(n) = n*sumdivmult(n, d, issquarefree(d)/d); \\ A001615
%o A306379 a(n) = sumdiv(n, d, f(d) * f(n/d)); \\ _Michel Marcus_, Feb 11 2019
%Y A306379 Cf. A001615.
%K A306379 nonn,mult,easy
%O A306379 1,2
%A A306379 _Torlach Rush_, Feb 11 2019