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%I #76 Apr 20 2024 10:17:41
%S 0,1,1,3,1,7,1,7,4,9,1,19,1,11,10,15,1,25,1,25,12,15,1,43,6,17,13,31,
%T 1,54,1,31,16,21,14,67,1,23,18,57,1,68,1,43,37,27,1,91,8,49,22,49,1,
%U 79,18,71,24,33,1,142,1,35,45,63,20,96,1,61,28,90,1,151,1,41,55
%N Dirichlet convolution of the identity function with omega.
%C a(n) = omega(n) = 1 iff n is prime.
%C a(n) = A323600(n) = 1 iff n is prime.
%C a(n) = A323600(n) - 1 = 1 iff n is the square of a prime.
%C a(n) = A323600(n) - 2 = 2 iff n is a squarefree semiprime.
%C a(n) = A323600(n) - (p + 2) if n is the cube of a prime p.
%H Antti Karttunen, <a href="/A323599/b323599.txt">Table of n, a(n) for n = 1..16384</a>
%H Antti Karttunen, <a href="/A323599/a323599.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F a(n) = Sum_{d|n} d * A001221(n/d).
%F a(n) = Sum_{p|n} sigma(n/p) where p is prime and sigma(n) = A000203(n). - _Ridouane Oudra_, Apr 28 2019
%F a(n) = Sum_{d|n} A069359(d), a(n) = A276085(A329380(n)). - _Antti Karttunen_, Nov 12 2019
%F From _Torlach Rush_, Mar 23 2024: (Start)
%F For p in primes: (Start)
%F a(p^(k+1)) = a(p^k) + p^k, k >= 0.
%F a(p^2) = p + 1.
%F (End)
%F a(2^k) = 2^k - 1, k >= 0.
%F (End)
%p with(numtheory):
%p a:= n-> add(d*nops(factorset(n/d)), d=divisors(n)):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jan 28 2019
%t Table[DivisorSum[n, # PrimeNu[n/#] &], {n, 75}] (* _Michael De Vlieger_, Jan 27 2019 *)
%o (PARI) a(n) = sumdiv(n, d, d*omega(n/d)); \\ _Michel Marcus_, Jan 22 2019
%Y Cf. A001221, A180253, A276085, A319684, A323600, A329347, A329380.
%Y Inverse Möbius transform of A069359.
%K nonn,changed
%O 1,4
%A _Torlach Rush_, Jan 18 2019
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