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A377517 The sum of the divisors of n that are terms in A276078.

%I A377517 #8 Oct 31 2024 01:11:21
%S A377517 1,3,4,3,6,12,8,3,13,18,12,12,14,24,24,3,18,39,20,18,32,36,24,12,31,
%T A377517 42,13,24,30,72,32,3,48,54,48,39,38,60,56,18,42,96,44,36,78,72,48,12,
%U A377517 57,93,72,42,54,39,72,24,80,90,60,72,62,96,104,3,84,144,68,54
%N A377517 The sum of the divisors of n that are terms in A276078.
%C A377517 First differs from A046897 at n = 27 = 3^3: a(27) = 13, while A046897(27) = 40.
%C A377517 The number of these divisors is A377516(n), and the largest of them is A377515(n).
%H A377517 Amiram Eldar, <a href="/A377517/b377517.txt">Table of n, a(n) for n = 1..10000</a>
%F A377517 a(n) = A000203(A377515(n)).
%F A377517 Multiplicative with a(p^e) = (p^(min(pi(p), e)+1) - 1)/(p - 1), where pi(n) = A000720(n).
%F A377517 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((pi(p)+1)*s) - p^(pi(p)+1))/p^((pi(p)+1)*s).
%F A377517 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^(pi(p)+1)) = 1.18603586369737251334... .
%t A377517 f[p_, e_] := (p^(Min[PrimePi[p], e] + 1) - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A377517 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(primepi(f[i,1]), f[i,2]) + 1) - 1)/(f[i,1] - 1));}
%Y A377517 Cf. A000203, A000720, A013661, A046897, A276078, A377515, A377516, A377520.
%K A377517 nonn,easy,mult
%O A377517 1,2
%A A377517 _Amiram Eldar_, Oct 30 2024