A392448 The sum of the exponential divisors of n that are numbers whose prime factorization exponents are all powers of 2 (A138302).

1, 2, 3, 6, 5, 6, 7, 2, 12, 10, 11, 18, 13, 14, 15, 22, 17, 24, 19, 30, 21, 22, 23, 6, 30, 26, 3, 42, 29, 30, 31, 2, 33, 34, 35, 72, 37, 38, 39, 10, 41, 42, 43, 66, 60, 46, 47, 66, 56, 60, 51, 78, 53, 6, 55, 14, 57, 58, 59, 90, 61, 62, 84, 6, 65, 66, 67, 102, 69, 70

Offset

1,2

Comments

The number of these divisors is A392447(n), and the largest of them is A365296(n).

Links

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

Formula

Multiplicative with a(p^e) = Sum_{k=0..v_2(e)} p^(2^k), where v_2(e) = A007814(e) is the 2-adic valuation of e.

a(n) <= A051377(n), with equality if and only if n is an exponentially 2^n number (A138302).

a(n) >= A007947(n), with equality if and only if n is an exponentially odd number (A268335).

Sum_{k=1..n} a(k) ~ n^2 / 2.

Mathematica

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

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, sum(k = 0, valuation(f[i, 2], 2), f[i, 1]^(1 << k))); }

Crossrefs

Cf. A007947, A051377, A138302, A268335, A322791, A392447.

Sequence in context: A253413 A337355 A384556 * A093783 A096861 A118738

Adjacent sequences: A392445 A392446 A392447 * A392449 A392450 A392451

Keyword

nonn,mult,easy

Author

Amiram Eldar, Jan 13 2026

Status

approved