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A371242 The sum of the unitary divisors of n that are cubefree numbers (A004709).

%I A371242 #14 Jun 14 2025 12:10:39
%S A371242 1,3,4,5,6,12,8,1,10,18,12,20,14,24,24,1,18,30,20,30,32,36,24,4,26,42,
%T A371242 1,40,30,72,32,1,48,54,48,50,38,60,56,6,42,96,44,60,60,72,48,4,50,78,
%U A371242 72,70,54,3,72,8,80,90,60,120,62,96,80,1,84,144,68,90,96
%N A371242 The sum of the unitary divisors of n that are cubefree numbers (A004709).
%C A371242 The number of these divisors is A365498(n).
%D A371242 D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.
%H A371242 Amiram Eldar, <a href="/A371242/b371242.txt">Table of n, a(n) for n = 1..10000</a>
%H A371242 Francesco Pappalardi, <a href="http://www.mat.uniroma3.it/users/pappa/papers/allahabad2003.pdf">A survey on k-freeness</a>, Number Theory, Ramanujan Math. Soc. Lect. Notes Ser., Vol. 1 (2003), pp. 71-88.
%F A371242 Multiplicative with a(p^e) = p^e + 1 for e <= 2, and a(p^e) = 1 for e >= 3.
%F A371242 a(n) = 1 if and only if n is cubefull (A036966).
%F A371242 a(n) = A000203(n) if and only if n is squarefree (A005117).
%F A371242 a(n) <= A034448(n), with equality if and only if n is cubefree.
%F A371242 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1) - 1/p^(3*s-2)).
%F A371242 Sum_{j=1..n} a(j) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^3 + 1/(p^2 + p)) = 1.16545286600957717104.... .
%F A371242 In general, the formula above holds for the sum of unitary divisors of n that are k-free numbers (k >= 2) with c = Product_{p prime} (1 - 1/p^k + 1/(p^2 + p)) (Suryanarayana, 1969). If k = 2 then c = A065465. In the limit when k -> oo, c = A306633.
%t A371242 f[p_, e_] := If[e < 3, p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A371242 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < 3, f[i, 1]^f[i, 2] + 1, 1));}
%Y A371242 Cf. A000203, A004709, A005117, A034448, A036966, A065465, A092261 (squarefree analog), A306633, A365498.
%K A371242 nonn,easy,mult
%O A371242 1,2
%A A371242 _Amiram Eldar_, Mar 16 2024