OFFSET
1,2
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{k=1..n} sigma(gcd(n,k)) * sigma(n/gcd(n,k)).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p^e) = (p^(2*e+3) - (e+1)*(p^2-1)*p^e - p)/((p-1)^2*(p+1)).
MATHEMATICA
Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[1, n/d], {d, Divisors[n]}], {n, 52}]
Table[Sum[DivisorSigma[1, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 52}]
f[p_, e_] := (p^(2*e + 3) - (e + 1)*(p^2 - 1)*p^e - p)/((p - 1)^2*(p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 12 2022 *)
PROG
(PARI) a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*sigma(n/d)); \\ Michel Marcus, Feb 17 2021
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Ilya Gutkovskiy, Feb 16 2021
STATUS
approved