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A348219 a(n) = tau(n) - omega(n) + n * Sum_{p|n, p prime} 1/p.

%I A348219 #21 Jul 21 2025 22:20:08
%S A348219 1,2,2,4,2,7,2,7,5,9,2,14,2,11,10,12,2,19,2,18,12,15,2,26,7,17,12,22,
%T A348219 2,36,2,21,16,21,14,37,2,23,18,34,2,46,2,30,28,27,2,48,9,39,22,34,2,
%U A348219 51,18,42,24,33,2,71,2,35,34,38,20,66,2,42,28,64,2,70,2,41,44,46
%N A348219 a(n) = tau(n) - omega(n) + n * Sum_{p|n, p prime} 1/p.
%C A348219 For each divisor d of n, add n/d if d is prime, otherwise add 1. For example, a(9) = 5 can be found using its divisors 1,3,9 to get 1 + 9/3 + 1 = 5.
%C A348219 If p is prime, then a(p) = 2 since we have a(p) = tau(p) - omega(p) + p/p = 2 - 1 + 1 = 2.
%H A348219 Antti Karttunen, <a href="/A348219/b348219.txt">Table of n, a(n) for n = 1..20000</a>
%F A348219 a(n) = Sum_{d|n} (n/d)^c(d), where c is the prime characteristic (A010051).
%F A348219 a(n) = A000005(n) - A001221(n) + A069359(n).
%F A348219 a(prime(n)) = 2.
%F A348219 From _Wesley Ivan Hurt_, Jul 21 2025: (Start)
%F A348219 a(n) = Sum_{d|n} (c(d) + phi(d)*omega(n/d)), where c = A005171.
%F A348219 a(n) = A007503(n) - A386438(n). (End)
%o A348219 (PARI) A348219(n) = sumdiv(n,d,(n/d)^isprime(d)); \\ _Antti Karttunen_, Nov 11 2021
%o A348219 (PARI)
%o A348219 A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
%o A348219 A348219(n) = (A069359(n)+numdiv(n)-omega(n)); \\ _Antti Karttunen_, Nov 11 2021
%Y A348219 Cf. A000005 (tau), A000010 (phi), A001221 (omega), A005171, A007503, A010051, A069359, A348203, A386438.
%K A348219 nonn
%O A348219 1,2
%A A348219 _Wesley Ivan Hurt_, Oct 07 2021