OFFSET
1,2
COMMENTS
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
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)));
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Ilya Gutkovskiy, Aug 29 2019
STATUS
approved