A214408 Abundant numbers for which the abundance is not a divisor.

30, 36, 42, 48, 54, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 102, 108, 112, 114, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 198, 200, 204, 208, 210, 216, 220, 222, 228, 240, 246, 252, 258, 260, 264, 270, 272, 276, 280, 282

Offset: 1

Author

Alonso del Arte, Jul 16 2012

Keywords

• nonn

Comments

The abundance of a number is sigma(n) - 2n (A033880).

Most of these numbers are pseudoperfect (A005835), but more than one proper divisor is left out of the sum.

The first odd term is 945, the second is 1575. The smallest odd abundant number not in this sequence is 173369889, found by Donovan Johnson. Peter J. C. Moses has verified all other odd abundant numbers up to 1.4 * 10^19 have an abundance that is not a proper divisor.

Almost all multiples of 6 are in this sequence. Given a prime p > 3, the abundance of 6p works out to 12, but 6p is not divisible by 4, though it is by 2 and by 3. The abundance of 12p is 16p + 28, and clearly that is not a divisor of 12p. Multiples of 6 with more prime factors will have abundances that are greater than the largest proper divisor by greater margins still.

Examples

			The abundance of 36 is 19, but 19 is not a divisor of 36, hence 36 is in the sequence.
The abundance of 40 is 10, which is a divisor of 40, hence 40 is not in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Crossrefs

Cf. A033880, A005835, A181595, A006037 (weird numbers, a subset).

Programs

Maple

filter:= proc(n) local b; b:= numtheory:-sigma(n) - 2*n; b > 0 and n mod b <> 0 end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 27 2016

Mathematica

Select[A005101, Not[MemberQ[Divisors[#], DivisorSigma[1, #] - 2#]] &]
anumQ[n_]:=Module[{a=DivisorSigma[1,n]-2n},a>0&&!Divisible[n,a]]; Select[Range[300],anumQ] (* Harvey P. Dale, Dec 23 2016 *)

PARI

is(n) = {my(ab = sigma(n) - 2*n); ab > 0 && n % ab;} \\ Amiram Eldar, Apr 07 2024

Extensions

Corrected by Harvey P. Dale, Dec 23 2016