A385139 The sum of divisors d of n such that n/d has exponents in its prime factorization that are all powers of 2 (A138302).

1, 3, 4, 7, 6, 12, 8, 14, 13, 18, 12, 28, 14, 24, 24, 29, 18, 39, 20, 42, 32, 36, 24, 56, 31, 42, 39, 56, 30, 72, 32, 58, 48, 54, 48, 91, 38, 60, 56, 84, 42, 96, 44, 84, 78, 72, 48, 116, 57, 93, 72, 98, 54, 117, 72, 112, 80, 90, 60, 168, 62, 96, 104, 116, 84, 144

Offset: 1

Author

Amiram Eldar, Jun 19 2025

Keywords

• nonn
• mult
• easy

Links

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

Crossrefs

Cf. A004709, A138302, A353900.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), this sequence (exponentially 2^n).

Programs

Mathematica

f[p_, e_] := p^e + Sum[p^(e - 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~, f[i, 1]^f[i, 2] + sum(k = 0, logint(f[i, 2], 2), f[i, 1]^(f[i, 2]-2^k)));}

Formula

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

a(n) <= A000203(n), with equality if and only if n is cubefree (A004709).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + (1-1/p)*(Sum_{k>=1} (Sum_{j=0..floor(log_2(k))} 1/p^(k+2^j)))) = 1.62194750148969761827... .