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A344221
a(n) = Sum_{k=1..n} tau(gcd(k,n)^3), where tau(n) is the number of divisors of n.
6
1, 5, 6, 13, 8, 30, 10, 29, 21, 40, 14, 78, 16, 50, 48, 61, 20, 105, 22, 104, 60, 70, 26, 174, 43, 80, 66, 130, 32, 240, 34, 125, 84, 100, 80, 273, 40, 110, 96, 232, 44, 300, 46, 182, 168, 130, 50, 366, 73, 215, 120, 208, 56, 330, 112, 290, 132, 160, 62, 624, 64, 170, 210, 253, 128, 420
OFFSET
1,2
LINKS
FORMULA
a(n) = Sum_{d|n} phi(n/d) * tau(d^3).
a(n) = n * Sum_{d|n} 3^omega(d) / d.
If p is prime, a(p) = 3 + p.
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = (p^e*(p + 2) - 3)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 2/p^2) = 1.8019184198... . (End)
MATHEMATICA
Table[Sum[DivisorSigma[0, GCD[k, n]^3], {k, n}], {n, 100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
f[p_, e_] := (p^e*(p + 2) - 3)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)
PROG
(PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n)^3));
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^3));
(PARI) a(n) = n*sumdiv(n, d, 3^omega(d)/d);
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 12 2021
STATUS
approved