A306379 Dirichlet convolution of psi(n) with itself.

1, 6, 8, 21, 12, 48, 16, 60, 40, 72, 24, 168, 28, 96, 96, 156, 36, 240, 40, 252, 128, 144, 48, 480, 96, 168, 168, 336, 60, 576, 64, 384, 192, 216, 192, 840, 76, 240, 224, 720, 84, 768, 88, 504, 480, 288, 96, 1248, 176, 576, 288, 588, 108, 1008, 288, 960, 320

Offset: 1

Author

Torlach Rush, Feb 11 2019

Keywords

• nonn
• mult
• easy

Comments

For n>1, a(n)>=2*n+2 with equality iff n is prime. - Robert Israel, Feb 28 2019

Sum_{k>=1} 1/a(k) diverges. - Vaclav Kotesovec, Sep 20 2020

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Dedekind Function.
• Wikipedia, Dedekind psi function.

Crossrefs

Cf. A001615.

Programs

Maple

psi:= proc(n) local p; option remember;  n*mul(1+1/p, p = numtheory:-factorset(n)): end proc:
f:= proc(n) local d;
   add(psi(d)*psi(n/d),d = numtheory:-divisors(n))
end proc:
map(f, [$1..100]); # Robert Israel, Feb 28 2019

Mathematica

psi[n_] := n Times @@ (1+1/FactorInteger[n][[All, 1]]); psi[1] = 1;
a[n_] := Sum[psi[d] psi[n/d], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Oct 16 2020 *)
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 *)

PARI

f(n) = n*sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = sumdiv(n, d, f(d) * f(n/d)); \\ Michel Marcus, Feb 11 2019

Formula

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

From Jianing Song, Apr 28 2019: (Start)

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

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

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