A382788 The sum of divisors of n that are numbers whose number of divisors is a power of 2.

1, 3, 4, 3, 6, 12, 8, 11, 4, 18, 12, 12, 14, 24, 24, 11, 18, 12, 20, 18, 32, 36, 24, 44, 6, 42, 31, 24, 30, 72, 32, 11, 48, 54, 48, 12, 38, 60, 56, 66, 42, 96, 44, 36, 24, 72, 48, 44, 8, 18, 72, 42, 54, 93, 72, 88, 80, 90, 60, 72, 62, 96, 32, 11, 84, 144, 68, 54

Offset: 1

Author

Amiram Eldar, Apr 05 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A033634 at n = 32.

The sum of the terms of A036537 that divide n.

The number of these divisors is A372380(n) and the largest of them is A372379(n).

Links

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

Crossrefs

Cf. A000523, A033634, A036537, A372379, A353900, A372380.

Programs

Mathematica

f[p_, e_] := Sum[p^(2^k-1), {k, 0, Floor[Log2[e + 1]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, sum(k = 0, exponent(e[i]+1), p[i]^(2^k-1)));

Formula

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + Sum_{k>=1} a(p^k)/p^(2*k)) = 1.13143029377358401678... .