A390370 The sum of the infinitary divisors of n that are exponentially odd numbers (A268335).

1, 3, 4, 1, 6, 12, 8, 11, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 44, 1, 42, 31, 8, 30, 72, 32, 35, 48, 54, 48, 1, 38, 60, 56, 66, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 93, 72, 88, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72

Offset

1,2

Comments

First differs from A368471 at n = 32.

The number of these divisors is A363825(n), and the largest of them is A350389(n).

Links

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

Formula

Multiplicative with a(p^e) = 1 if e is even, and a(p^e) = 1 + p * isigma(p^(e-1)) if e is odd, where isigma(n) = A049417(n).

a(n) = A049417(n) if and only if n is exponentially odd.

a(n) = 1 if and only if n is a square.

Mathematica

f[p_, e_] := If[EvenQ[e], 1, 1 + p * Times @@ Flatten[(1 + p^(2^(-1 + Position[Reverse@ IntegerDigits[e - 1, 2], _?(# == 1 &)])))]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n), d); prod(i = 1, #f~, if(f[i, 2] % 2, d = binary(f[i, 2] - 1); 1 + f[i, 1] * prod(k = 1, #d, if(d[k], 1 + f[i, 1]^(2^(-k + #d)), 1)), 1)); }

Crossrefs

Cf. A049417, A077609, A268335, A350389, A358346, A363825, A368471, A384558, A384559, A390372.

Sequence in context: A358346 A348733 A388994 * A368471 A390372 A351569

Adjacent sequences: A390367 A390368 A390369 * A390371 A390372 A390373

Keyword

nonn,easy,mult

Author

Amiram Eldar, Nov 03 2025

Status

approved