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A306379 Dirichlet convolution of psi(n) with itself. 2
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 (list; graph; refs; listen; history; text; internal format)
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
Eric Weisstein's World of Mathematics, Dedekind 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
KEYWORD
nonn,mult,easy
AUTHOR
Torlach Rush, Feb 11 2019
STATUS
approved

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Last modified July 19 17:03 EDT 2024. Contains 374410 sequences. (Running on oeis4.)