A390264 Multiplicative with a(p^e) = e*p - 1.

1, 1, 2, 3, 4, 2, 6, 5, 5, 4, 10, 6, 12, 6, 8, 7, 16, 5, 18, 12, 12, 10, 22, 10, 9, 12, 8, 18, 28, 8, 30, 9, 20, 16, 24, 15, 36, 18, 24, 20, 40, 12, 42, 30, 20, 22, 46, 14, 13, 9, 32, 36, 52, 8, 40, 30, 36, 28, 58, 24, 60, 30, 30, 11, 48, 20, 66, 48, 44, 24, 70

Offset

1,3

Links

Aloe Poliszuk, Table of n, a(n) for n = 1..10000

Formula

a(n) = (-1)^omega(n) * Sum_{d|n} rad(d)*(-1)^omega(d) where omega = A001221, rad = A007947.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(2)^2 * Product_{p prime} (1 - 4/p^2 + 3/p^3 + 2/p^4 - 2/p^5) = 0.2888609151222355343006... . - Amiram Eldar, Oct 31 2025

a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*sigma(d)*tau(n/d), where omega = A001221. - Ridouane Oudra, Jun 25 2026

Example

a(6) = a(2^1)*a(3^1) = (2-1)*(3-1) = 2.

Mathematica

f[p_, e_] := e*p - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 31 2025 *)

Prog

(PARI) a(n) = my(fac = factorint(n)); prod(X=1, #fac[, 1], fac[X, 1]*fac[X, 2] - 1);
(PARI)
rad(n) = factorback(factorint(n)[, 1]); \\ from A007947
a(n) = sumdiv(n, d, rad(d)*(-1)^(omega(n)+omega(d)));

Crossrefs

Cf. A001221 (omega), A007947 (rad), A076479, A000005, A008683, A000203.

Related arithmetic functions: A000026, A191750.

Sequence in context: A109746 A384247 A286365 * A345061 A061020 A206369

Adjacent sequences: A390261 A390262 A390263 * A390265 A390266 A390267

Keyword

nonn,mult,easy

Author

Aloe Poliszuk, Oct 30 2025

Status

approved