|
%I #7 Jun 14 2021 12:26:34
%S 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 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 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 a(n) = Sum_{d|n} d^[Omega(d) = 2], where [ ] is the Iverson bracket.
%C 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.
%F a(p) = Sum_{d|p} d^[Omega(d) = 2] = 1^0 + p^0 = 2, for primes p.
%e a(n) = Sum_{d|12} d^[Omega(d) = 2] = 1^0 + 2^0 + 3^0 + 4^1 + 6^1 + 12^0 = 14.
%t Table[Sum[k^KroneckerDelta[PrimeOmega[k], 2] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
%o (PARI) a(n) = sumdiv(n, d, if (bigomega(d)==2, d, 1)); \\ _Michel Marcus_, Jun 13 2021
%Y Cf. A001222 (Omega).
%K nonn
%O 1,2
%A _Wesley Ivan Hurt_, Jun 13 2021
|
