A192268 Anti-abundant numbers.

7, 10, 11, 12, 13, 14, 15, 17, 18, 20, 21, 22, 23, 25, 27, 28, 30, 31, 32, 33, 35, 37, 38, 39, 40, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 55, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113

Offset: 1

Author

Paolo P. Lava, Jun 28 2011

Keywords

• nonn
• easy

Comments

An anti-abundant number is a number n for which sigma*(n) > n, where sigma*(n) is the sum of the anti-divisors of n. Like A005101 but using anti-divisors.

Examples

			25 is anti-abundant because its anti-divisors are 2, 3, 7, 10, 17 and their sum is 39 > 25.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Paolo P. Lava)

Crossrefs

Cf. A066417, A005101, A066272, A192267.

Programs

Maple

isA192268 := proc(n) A066417(n) > n ; end proc:
for n from 1 to 500 do if isA192268(n) then printf("%d,",n); end if; end do: # R. J. Mathar, Jul 04 2011

Mathematica

antiAbQ[n_] := Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]] > n; Select[Range[120], antiAbQ] (* _Amiram Eldar, Jan 13 2022 after Michael De Vlieger at A066417 *)

Python

from itertools import count, islice
from sympy import divisor_sigma, multiplicity
def A192268gen(): return filter(lambda n:divisor_sigma(2*n-1)+divisor_sigma(2*n+1)+divisor_sigma(n//2**(k:=multiplicity(2,n)))*2**(k+1)-7*n-2 > 0,count(2))
A192268_list = list(islice(A192268gen(),16)) # Chai Wah Wu, Dec 23 2021

Formula

A000027 = A073930 UNION A192267 UNION {this set}.