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A362854 The sum of the divisors of n that are both bi-unitary and exponential. 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
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
Sequence in context: A266646 A364280 A361810 * A102455 A073678 A210947
KEYWORD
nonn,mult
AUTHOR
Amiram Eldar, May 05 2023
STATUS
approved

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Last modified June 24 03:36 EDT 2024. Contains 373661 sequences. (Running on oeis4.)