This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A214408 #32 Apr 07 2024 09:09:49
%S A214408 30,36,42,48,54,60,66,70,72,78,80,84,90,96,100,102,108,112,114,126,
%T A214408 132,138,140,144,150,156,160,162,168,174,176,180,186,192,198,200,204,
%U A214408 208,210,216,220,222,228,240,246,252,258,260,264,270,272,276,280,282
%N A214408 Abundant numbers for which the abundance is not a divisor.
%C A214408 The abundance of a number is sigma(n) - 2n (A033880).
%C A214408 Most of these numbers are pseudoperfect (A005835), but more than one proper divisor is left out of the sum.
%C A214408 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.
%C A214408 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.
%H A214408 Amiram Eldar, <a href="/A214408/b214408.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%e A214408 The abundance of 36 is 19, but 19 is not a divisor of 36, hence 36 is in the sequence.
%e A214408 The abundance of 40 is 10, which is a divisor of 40, hence 40 is not in the sequence.
%p A214408 filter:= proc(n) local b; b:= numtheory:-sigma(n) - 2*n; b > 0 and n mod b <> 0 end proc:
%p A214408 select(filter, [$1..1000]); # _Robert Israel_, Dec 27 2016
%t A214408 Select[A005101, Not[MemberQ[Divisors[#], DivisorSigma[1, #] - 2#]] &]
%t A214408 anumQ[n_]:=Module[{a=DivisorSigma[1,n]-2n},a>0&&!Divisible[n,a]]; Select[Range[300],anumQ] (* _Harvey P. Dale_, Dec 23 2016 *)
%o A214408 (PARI) is(n) = {my(ab = sigma(n) - 2*n); ab > 0 && n % ab;} \\ _Amiram Eldar_, Apr 07 2024
%Y A214408 Cf. A033880, A005835, A181595, A006037 (weird numbers, a subset).
%K A214408 nonn
%O A214408 1,1
%A A214408 _Alonso del Arte_, Jul 16 2012
%E A214408 Corrected by _Harvey P. Dale_, Dec 23 2016