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A322577
a(n) = Sum_{d|n} psi(n/d) * phi(d).
8
1, 4, 6, 11, 10, 24, 14, 28, 26, 40, 22, 66, 26, 56, 60, 68, 34, 104, 38, 110, 84, 88, 46, 168, 74, 104, 102, 154, 58, 240, 62, 160, 132, 136, 140, 286, 74, 152, 156, 280, 82, 336, 86, 242, 260, 184, 94, 408, 146, 296, 204, 286, 106, 408, 220, 392, 228, 232, 118, 660
OFFSET
1,2
COMMENTS
Dirichlet convolution of Dedekind psi function (A001615) with Euler totient function (A000010).
Dirichlet convolution of A008966 with A018804.
Dirichlet convolution of A038040 with A271102.
LINKS
Eric Weisstein's World of Mathematics, Dedekind Function.
Eric Weisstein's World of Mathematics, Totient Function.
FORMULA
Dirichlet g.f.: zeta(s-1)^2 / zeta(2*s).
a(p) = 2*p, where p is prime.
Sum_{k=1..n} a(k) ~ 45*n^2*(2*Pi^4*log(n) - Pi^4 + 4*gamma*Pi^4 - 360*zeta'(4)) / (2*Pi^8), where gamma is the Euler-Mascheroni constant A001620 and for zeta'(4) see A261506. - Vaclav Kotesovec, Aug 31 2019
a(p^k) = (k+1)*p^k - (k-1)*p^(k-2) where p is prime. - Robert Israel, Sep 01 2019
a(n) = Sum_{k=1..n} psi(gcd(n,k)). - Ridouane Oudra, Nov 29 2019
a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
MAPLE
f:= proc(n) local t;
mul((t[2]+1)*t[1]^t[2] - (t[2]-1)*t[1]^(t[2]-2), t = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Sep 01 2019
MATHEMATICA
Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, n/d] EulerPhi[d], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := (e + 1)*p^e - (e - 1)*p^(e - 2); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)
PROG
(PARI) seq(n) = {dirmul(vector(n, n, eulerphi(n)), vector(n, n, n * sumdivmult(n, d, issquarefree(d)/d)))} \\ Andrew Howroyd, Aug 29 2019
(PARI)
A001615(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A322577(n) = sumdiv(n, d, A001615(n/d)*eulerphi(d)); \\ Antti Karttunen, Apr 03 2022
CROSSREFS
Cf. A327251 (inverse Möbius transform), A347092 (Dirichlet inverse), A347093 (sum with it), A347135.
Sequence in context: A273103 A187215 A365647 * A013018 A031271 A369329
KEYWORD
nonn,mult,easy
AUTHOR
Ilya Gutkovskiy, Aug 29 2019
STATUS
approved