A328181 a(n) = (-1)^(bigomega(n) - omega(n)) * Sum_{d|n} (-1)^(bigomega(d) - omega(d)) * d.

1, 3, 4, 1, 6, 12, 8, 7, 5, 18, 12, 4, 14, 24, 24, 9, 18, 15, 20, 6, 32, 36, 24, 28, 19, 42, 22, 8, 30, 72, 32, 23, 48, 54, 48, 5, 38, 60, 56, 42, 42, 96, 44, 12, 30, 72, 48, 36, 41, 57, 72, 14, 54, 66, 72, 56, 80, 90, 60, 24, 62, 96, 40, 41, 84, 144, 68, 18, 96, 144

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Eric Weisstein's World of Mathematics, Prime Factor.

Formula

a(p) = p + 1, where p is prime.

Multiplicative with a(p^e) = (p^(e+1) - (-1)^e*(2*p+1))/(p+1). - Amiram Eldar, Dec 02 2020

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/30) * Product_{p prime} (1 + 2/p^2 - 2/p^3) = 0.5507877576... . - Amiram Eldar, Nov 06 2022

Mathematica

a[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]) Sum[(-1)^(PrimeOmega[d] - PrimeNu[d]) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
f[p_, e_] := (p^(e+1) - (-1)^e *(2*p+1))/(p+1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

Prog

(PARI) a(n) = (-1)^(bigomega(n)-omega(n))*sumdiv(n, d, (-1)^(bigomega(d)-omega(d))*d); \\ Michel Marcus, Oct 06 2019

Crossrefs

Cf. A001221, A001222, A008836, A046660, A049060, A055076, A061020, A076479, A162511.

Sequence in context: A348984 A323159 A347090 * A358346 A348733 A368471

Adjacent sequences: A328178 A328179 A328180 * A328182 A328183 A328184

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Oct 06 2019

Status

approved