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

Author

Amiram Eldar, May 28 2023

Keywords

• nonn
• mult

Comments

First differs from A363334 at n = 16.

The number of these divisors is A363329(n).

Examples

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

Links

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

Crossrefs

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

Programs

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]

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);}

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... .