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A349215 a(n) = Sum_{d|n} d^c(d), where c is the prime characteristic (A010051).

%I A349215 #19 Nov 28 2021 11:44:23
%S A349215 1,3,4,4,6,7,8,5,5,9,12,9,14,11,10,6,18,9,20,11,12,15,24,11,7,17,6,13,
%T A349215 30,15,32,7,16,21,14,12,38,23,18,13,42,17,44,17,12,27,48,13,9,11,22,
%U A349215 19,54,11,18,15,24,33,60,19,62,35,14,8,20,21,68,23,28,19,72,15,74,41,12
%N A349215 a(n) = Sum_{d|n} d^c(d), where c is the prime characteristic (A010051).
%C A349215 For each divisor d of n, add d if d is prime, otherwise add 1. For example, a(6) = 1 + 2 + 3 + 1 = 7.
%H A349215 Antti Karttunen, <a href="/A349215/b349215.txt">Table of n, a(n) for n = 1..20000</a>
%F A349215 a(n) = A000005(n)+A008472(n)-A001221(n). - _Chai Wah Wu_, Nov 11 2021
%F A349215 a(p) = p+1 for primes p. - _Wesley Ivan Hurt_, Nov 28 2021
%t A349215 a[n_] := DivisorSum[n, #^Boole[PrimeQ[#]] &]; Array[a, 75] (* _Amiram Eldar_, Nov 11 2021 *)
%o A349215 (PARI) a(n) = sumdiv(n, d, if (isprime(d), d, 1)); \\ _Michel Marcus_, Nov 11 2021
%o A349215 (Python)
%o A349215 from math import prod
%o A349215 from sympy import factorint
%o A349215 def A349215(n):
%o A349215     fs = factorint(n)
%o A349215     return sum(a-1 for a in fs.keys())+prod(1+d for d in fs.values()) # _Chai Wah Wu_, Nov 11 2021
%Y A349215 Inverse Moebius transform of A089026.
%Y A349215 Cf. A010051, A000005 (tau), A008472 (sopf), A001221 (omega).
%K A349215 nonn
%O A349215 1,2
%A A349215 _Wesley Ivan Hurt_, Nov 10 2021