A306379 - OEIS

%I #33 Oct 22 2020 06:36:16

%S 1,6,8,21,12,48,16,60,40,72,24,168,28,96,96,156,36,240,40,252,128,144,

%T 48,480,96,168,168,336,60,576,64,384,192,216,192,840,76,240,224,720,

%U 84,768,88,504,480,288,96,1248,176,576,288,588,108,1008,288,960,320

%N Dirichlet convolution of psi(n) with itself.

%C For n>1, a(n)>=2*n+2 with equality iff n is prime. - _Robert Israel_, Feb 28 2019

%C Sum_{k>=1} 1/a(k) diverges. - _Vaclav Kotesovec_, Sep 20 2020

%H Robert Israel, <a href="/A306379/b306379.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DedekindFunction.html">Dedekind Function</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dedekind_psi_function">Dedekind psi function</a>.

%F a(n) = Sum_{d|n} psi(d) * psi(n/d).

%F From _Jianing Song_, Apr 28 2019: (Start)

%F Multiplicative with a(p^e) = (e-1)*(p+1)^2*p^(e-2) + 2*(p+1)*p^(e-1).

%F Dirichlet g.f.: (zeta(s) * zeta(s-1) / zeta(2*s))^2. (End)

%F 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 psi:= proc(n) local p; option remember;  n*mul(1+1/p, p = numtheory:-factorset(n)): end proc:

%p f:= proc(n) local d;

%p    add(psi(d)*psi(n/d),d = numtheory:-divisors(n))

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Feb 28 2019

%t psi[n_] := n Times @@ (1+1/FactorInteger[n][[All, 1]]); psi[1] = 1;

%t a[n_] := Sum[psi[d] psi[n/d], {d, Divisors[n]}];

%t Array[a, 100] (* _Jean-François Alcover_, Oct 16 2020 *)

%t 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 (PARI) f(n) = n*sumdivmult(n, d, issquarefree(d)/d); \\ A001615

%o a(n) = sumdiv(n, d, f(d) * f(n/d)); \\ _Michel Marcus_, Feb 11 2019

%Y Cf. A001615.

%K nonn,mult,easy

%O 1,2

%A _Torlach Rush_, Feb 11 2019