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A353900 a(n) is the sum of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

%I A353900 #15 Nov 19 2022 04:47:37
%S A353900 1,3,4,7,6,12,8,7,13,18,12,28,14,24,24,23,18,39,20,42,32,36,24,28,31,
%T A353900 42,13,56,30,72,32,23,48,54,48,91,38,60,56,42,42,96,44,84,78,72,48,92,
%U A353900 57,93,72,98,54,39,72,56,80,90,60,168,62,96,104,23,84,144
%N A353900 a(n) is the sum of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).
%H A353900 Amiram Eldar, <a href="/A353900/b353900.txt">Table of n, a(n) for n = 1..10000</a>
%F A353900 Multiplicative with a(p^e) = 1 + Sum_{k=0..floor(log_2(e))} p^(2^k).
%F A353900 a(n) = A000203(n) if and only if n is cubefree (A004709).
%F A353900 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1-1/p)*(1 + Sum_{k>=1} (Sum_{j=0..floor(log_2(k))} p^(2^j)/p^(2*k)))) = 0.7176001667... . - _Amiram Eldar_, Nov 19 2022
%t A353900 f[p_, e_] := 1 + Sum[p^(2^k), {k, 0, Floor[Log2[e]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A353900 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sum(k = 0, logint(f[i,2], 2), f[i,1]^(2^k)));} \\ _Amiram Eldar_, Nov 19 2022
%Y A353900 Cf. A000203, A004709, A138302, A353898.
%Y A353900 Similar sequences: A034448, A048146, A051377, A188999.
%K A353900 nonn,mult
%O A353900 1,2
%A A353900 _Amiram Eldar_, May 10 2022