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