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

1,2

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.

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

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 *)

Prog

(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

Crossrefs

Cf. A001615.

Sequence in context: A096524 A083595 A064840 * A243932 A242504 A267023

Adjacent sequences: A306376 A306377 A306378 * A306380 A306381 A306382

Keyword

nonn,mult,easy

Author

Torlach Rush, Feb 11 2019

Status

approved