The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A361810 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A361810

a(n) is the sum of divisors of n that are both infinitary and exponential.
(history; published version)

#9 by Joerg Arndt at Mon Mar 27 03:38:17 EDT 2023

STATUS

reviewed

approved

#8 by Michel Marcus at Mon Mar 27 02:08:00 EDT 2023

STATUS

proposed

reviewed

#7 by Amiram Eldar at Mon Mar 27 01:55:29 EDT 2023

STATUS

editing

proposed

#6 by Amiram Eldar at Mon Mar 27 01:36:38 EDT 2023

LINKS

Amiram Eldar, <a href="/A361810/b361810.txt">Table of n, a(n) for n = 1..10000</a>

STATUS

approved

editing

#5 by Michael De Vlieger at Sat Mar 25 08:18:42 EDT 2023

STATUS

reviewed

approved

#4 by Joerg Arndt at Sat Mar 25 07:16:00 EDT 2023

STATUS

proposed

reviewed

#3 by Amiram Eldar at Sat Mar 25 04:38:45 EDT 2023

STATUS

editing

proposed

#2 by Amiram Eldar at Sat Mar 25 03:32:22 EDT 2023

NAME

allocated for Amiram Eldara(n) is the sum of divisors of n that are both infinitary and exponential.

DATA

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.

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.

KEYWORD

allocated

nonn,easy,mult

AUTHOR

Amiram Eldar, Mar 25 2023

STATUS

approved

editing

#1 by Amiram Eldar at Sat Mar 25 03:32:22 EDT 2023

NAME

allocated for Amiram Eldar

KEYWORD

allocated

STATUS

approved