A366903 The sum of exponentially odious divisors of n.

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, 68

Offset: 1

Author

Amiram Eldar, Oct 27 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A353900 at n = 128.

The number of these divisors is A366901(n) and the largest of them is A366905(n).

Links

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

Crossrefs

Cf. A004709, A000203, A366901, A366905.

Similar sequences: A353900, A365682, A366904.

Programs

Mathematica

f[p_, e_] := 1 + Total[p^Select[Range[e], OddQ[DigitCount[#, 2, 1]] &]]; 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 = 1, f[i, 2], (hammingweight(k)%2) * f[i, 1]^k));}

Formula

Multiplicative with a(p^e) = 1 + Sum_{k = 1..e, k is odious} p^k.

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

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1-1/p)*(1 + Sum_{k>=1} a(p^k)/p^(2*k)) = 0.721190607... .