A361810 a(n) is the sum of divisors of n that are both infinitary and exponential.

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

Offset

1,2

Comments

The number of these divisors is A359411(n).

The indices of records of a(n)/n are the primorials (A002110) cubed, i.e., 1 and the terms of A115964.

Links

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

Formula

Multiplicative with a(p^e) = Sum_{d|e, bitor(d, e) == e} p^d.

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

limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p)*(1 + Sum_{e>=1} Sum_{d|e, bitor(d, e) == e} p^(d-2*e))) = 0.51015879911178031024... .

Example

a(8) = 10 since 8 has 2 divisors that are both infinitary and exponential, 2 and 8, and 2 + 8 = 10.

Mathematica

f[p_, e_] := DivisorSum[e, p^# &, BitOr[#, e] == e &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) s(p, e) = sumdiv(e, d, p^d*(bitor(d, e) == e));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 1], f[i, 2])); }

Crossrefs

Cf. A002110, A049417, A051377, A082020, A115964, A138302, A359411.

Sequence in context: A266646 A364280 A374575 * A362854 A102455 A073678

Adjacent sequences: A361807 A361808 A361809 * A361811 A361812 A361813

Keyword

nonn,easy,mult

Author

Amiram Eldar, Mar 25 2023

Status

approved