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

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, 42, 13, 56, 30, 72, 32, 23, 48, 54, 48, 91, 38, 60, 56, 42, 42, 96, 44, 84, 78, 72, 48, 92, 57, 93, 72, 98, 54, 39, 72, 56, 80, 90, 60, 168, 62, 96, 104, 23, 84, 144

Offset: 1

Author

Amiram Eldar, May 10 2022

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A004709, A138302, A353898.

Similar sequences: A034448, A048146, A051377, A188999.

Programs

Mathematica

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]

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

Formula

Multiplicative with a(p^e) = 1 + Sum_{k=0..floor(log_2(e))} p^(2^k).

a(n) = A000203(n) if and only if n is cubefree (A004709).

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