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A230593 a(n) = n * Sum_{q|n} 1 / q, where q are noncomposite numbers (A008578) dividing n.

%I A230593 #18 Nov 12 2021 12:34:18
%S A230593 1,3,4,6,6,11,8,12,12,17,12,22,14,23,23,24,18,33,20,34,31,35,24,44,30,
%T A230593 41,36,46,30,61,32,48,47,53,47,66,38,59,55,68,42,83,44,70,69,71,48,88,
%U A230593 56,85,71,82,54,99,71,92,79,89,60,122,62,95,93,96,83,127
%N A230593 a(n) = n * Sum_{q|n} 1 / q, where q are noncomposite numbers (A008578) dividing n.
%H A230593 Antti Karttunen, <a href="/A230593/b230593.txt">Table of n, a(n) for n = 1..20000</a>
%F A230593 For n > 1, a(n) = n + n * Sum_(p|n) 1 / p, where p are primes dividing n.
%F A230593 a(n) = A069359(n) + n.
%F A230593 a(n) = Sum_{d|n}  A080339(d) * A000027(n/d).
%F A230593 a(n) = A080339(n) * A000027(n), where operation * denotes Dirichlet convolution, i.e. convolution of type: a(n) = Sum_{d|n} b(d) * c(n/d).
%F A230593 For p, q = distinct primes, a(p) = p + 1, a(pq) = pq - 1.
%F A230593 From _Antti Karttunen_, Nov 12 2021: (Start)
%F A230593 a(n) = A129283(n) - A329039(n).
%F A230593 a(A005117(n)) = A129283(A005117(n)), for all n >= 1.
%F A230593 (End)
%F A230593 For p prime, k>=1, a(p^k) = p^(k-1) * (p+1). - _Bernard Schott_, Nov 12 2021
%e A230593 For n = 6: a(6) = 6 * (1/1 + 1/2 + 1/3) = 11.
%t A230593 a[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 12 2021 *)
%o A230593 (PARI) A230593(n) = sumdiv(n,d,((1==d)||isprime(d))*(n/d)); \\ _Antti Karttunen_, Nov 12 2021
%Y A230593 Cf. A080339, A000027, A008578, A329039.
%Y A230593 Coincides with A129283 on squarefree numbers, A005117.
%K A230593 nonn
%O A230593 1,2
%A A230593 _Jaroslav Krizek_, Oct 25 2013