A361810 -id:A361810

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A362854

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn,mult

AUTHOR

Amiram Eldar, May 05 2023

STATUS

approved

A363331

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

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); }