A362854 The sum of the divisors of n that are both bi-unitary and exponential.

1, 2, 3, 4, 5, 6, 7, 10, 9, 10, 11, 12, 13, 14, 15, 18, 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, 54, 49, 50, 51, 52, 53, 60, 55, 70, 57, 58, 59, 60, 61, 62, 63, 70, 65, 66, 67, 68

Offset

1,2

Comments

The number of these divisors is A362852(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} p^d if e is odd, and (Sum_{d|e} p^d) - p^(e/2) if e is even.

a(n) >= n, with equality if and only if n is cubefree (A004709).

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, d != e/2}, p^(d-2*e))) = 0.5124353304539905... .

Example

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

Mathematica

f[p_, e_] := DivisorSum[e, p^# &] - If[OddQ[e], 0, p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

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

Crossrefs

Cf. A002110, A004709, A051377, A082020, A115964, A188999, A222266, A322791, A361810, A362852.

Sequence in context: A364280 A374575 A361810 * A102455 A073678 A210947

Adjacent sequences: A362851 A362852 A362853 * A362855 A362856 A362857

Keyword

nonn,mult

Author

Amiram Eldar, May 05 2023

Status

approved