A363331 a(n) is the sum of divisors of n that are both coreful and infinitary.

1, 2, 3, 4, 5, 6, 7, 14, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 42, 25, 26, 39, 28, 29, 30, 31, 50, 33, 34, 35, 36, 37, 38, 39, 70, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 78, 55, 98, 57, 58, 59, 60, 61, 62, 63, 84, 65, 66, 67, 68

Offset

1,2

Comments

First differs from A363334 at n = 16.

The number of these divisors is A363329(n).

Links

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

Formula

Multiplicative with a(p^e) = (Product_{k>=0} ((p^(2^k*(b(k)+1)) - 1)/(p^(2^k) - 1)) - 1, where e = Sum_{k >= 0} b(k) * 2^k is the binary representation of e.

a(n) >= n, with equality if and only if n is in A138302.

a(n) >= A361810(n), with equality if and only if n is in A138302.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p) * Sum_{k>=1} a(p^k)/p^(2*k)) = 0.53906337497505398777... .

Example

a(8) = 14 since 8 has 3 divisors that are both infinitary and coreful, 2, 4 and 8, and 2 + 4 + 8 = 14.

Mathematica

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

Prog

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

Crossrefs

Cf. A049417, A057723, A077609, A138302, A361810, A363329, A363334.

Sequence in context: A060810 A319723 A271377 * A363334 A066336 A076133

Adjacent sequences: A363328 A363329 A363330 * A363332 A363333 A363334

Keyword

nonn,mult

Author

Amiram Eldar, May 28 2023

Status

approved