A109729 Odd admirable numbers: such that sigma(n) = 2n + 2d for some d | n.

945, 4095, 6435, 7425, 8415, 8925, 9555, 26145, 28035, 30555, 31815, 32445, 43065, 46035, 78975, 80535, 81081, 103455, 129195, 182655, 191565, 261261, 279279, 351351, 354585, 355725, 371925, 403095, 411255, 430815, 437745, 442365, 458055

Offset

1,1

Comments

Equivalently: Odd n such that sigma(n)/2 - n is a positive divisor of n. (Negative and/or half-integer d = sigma(n)/2 - n, of which n could be a multiple, are excluded. Negative d correspond to deficient n, half-integer d to square n: the first example of an abundant n being a multiple of a half-integer d is n = 13167^2 = 173369889.) - M. F. Hasler, Jan 26 2020

First term not divisible by 3 is 19594563749886125. - Alexander Violette, May 26 2026, verified by David A. Corneth, May 26 2026

Links

Amiram Eldar, Table of n, a(n) for n = 1..746 (terms below 3*10^12)

Mathematica

Select[Range[1, 460000, 2], MemberQ[2#+2 Divisors[#], DivisorSigma[1, #]]&] (* Harvey P. Dale, Mar 23 2025 *)

Prog

(PARI) select( {is_A109729(n, s=sigma(n))=s>2*n&&n%(s/2-n)==0&&n%2&&!(s%2)}, [2*k-1|k<-[1..5e5\2]]) \\ M. F. Hasler, Jan 26 2020

Crossrefs

Cf. A111592 (admirable numbers).

Cf. A000203 (sigma: sum of divisors).

Sequence in context: A252184 A188263 A188342 * A294025 A389701 A275449

Adjacent sequences: A109726 A109727 A109728 * A109730 A109731 A109732

Keyword

nonn

Author

Jason Earls, Aug 09 2005

Extensions

Offset corrected by Amiram Eldar, Jun 22 2019

Name edited by M. F. Hasler, Jan 26 2020

Status

approved