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

Author

Jason Earls, Aug 09 2005

Keywords

• nonn

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

Links

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

Crossrefs

Cf. A111592 (admirable numbers).

Cf. A000203 (sigma: sum of divisors).

Programs

Mathematica

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

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

Extensions

Offset corrected by Amiram Eldar, Jun 22 2019

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