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A345315 a(n) = Sum_{d|n} d^[Omega(d) = 2], where [ ] is the Iverson bracket.

%I A345315 #17 Jul 23 2025 11:26:11
%S A345315 1,2,2,6,2,9,2,7,11,13,2,14,2,17,18,8,2,19,2,18,24,25,2,16,27,29,12,
%T A345315 22,2,36,2,9,36,37,38,25,2,41,42,20,2,46,2,30,28,49,2,18,51,39,54,34,
%U A345315 2,21,58,24,60,61,2,43,2,65,34,10,68,66,2,42,72,64,2,28,2,77,44,46,80
%N A345315 a(n) = Sum_{d|n} d^[Omega(d) = 2], where [ ] is the Iverson bracket.
%C A345315 For each divisor d of n, add d if d is semiprime, otherwise add 1. For example, the divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24, and the only semiprime divisors of 24 are 4 and 6, so a(24) = 1 + 1 + 1 + 4 + 6 + 1 + 1 + 1 = 16.
%C A345315 Inverse Möbius transform of 1 + (n - 1)*c(n), where c = A064911. - _Wesley Ivan Hurt_, Jul 22 2025
%H A345315 Antti Karttunen, <a href="/A345315/b345315.txt">Table of n, a(n) for n = 1..20000</a>
%F A345315 From _Wesley Ivan Hurt_, Jul 22 2025: (Start)
%F A345315 a(p^k) = k + p^2 for p prime and k >= 2, else k + 1 if k = 0 or 1.
%F A345315 a(n) = tau(n) + Sum_{d|n} (d - 1) * c(d), where c = A064911.
%F A345315 a(n) = A076290(n) + A000005(n) - A086971(n). (End)
%e A345315 a(12) = Sum_{d|12} d^[Omega(d) = 2] = 1^0 + 2^0 + 3^0 + 4^1 + 6^1 + 12^0 = 14.
%t A345315 Table[Sum[k^KroneckerDelta[PrimeOmega[k], 2] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
%o A345315 (PARI) a(n) = sumdiv(n, d, if (bigomega(d)==2, d, 1)); \\ _Michel Marcus_, Jun 13 2021
%Y A345315 Cf. A000005 (tau), A001222 (Omega), A001358 (semiprimes), A064911, A076290, A086971.
%K A345315 nonn
%O A345315 1,2
%A A345315 _Wesley Ivan Hurt_, Jun 13 2021