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%I #15 Nov 06 2022 03:16:45
%S 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,
%T 22,8,30,72,32,23,48,54,48,5,38,60,56,42,42,96,44,12,30,72,48,36,41,
%U 57,72,14,54,66,72,56,80,90,60,24,62,96,40,41,84,144,68,18,96,144
%N a(n) = (-1)^(bigomega(n) - omega(n)) * Sum_{d|n} (-1)^(bigomega(d) - omega(d)) * d.
%H Amiram Eldar, <a href="/A328181/b328181.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>.
%F a(p) = p + 1, where p is prime.
%F Multiplicative with a(p^e) = (p^(e+1) - (-1)^e*(2*p+1))/(p+1). - _Amiram Eldar_, Dec 02 2020
%F 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
%t a[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]) Sum[(-1)^(PrimeOmega[d] - PrimeNu[d]) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
%t 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 *)
%o (PARI) a(n) = (-1)^(bigomega(n)-omega(n))*sumdiv(n, d, (-1)^(bigomega(d)-omega(d))*d); \\ _Michel Marcus_, Oct 06 2019
%Y Cf. A001221, A001222, A008836, A046660, A049060, A055076, A061020, A076479, A162511.
%K nonn,mult
%O 1,2
%A _Ilya Gutkovskiy_, Oct 06 2019