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A275082 Primitive abundant numbers (having no abundant proper divisors) that have perfect proper divisors.

%I A275082 #14 Jun 26 2019 05:38:52
%S A275082 12,18,30,42,56,66,78,102,114,138,174,186,196,222,246,258,282,308,318,
%T A275082 354,364,366,402,426,438,474,476,498,532,534,582,606,618,642,644,654,
%U A275082 678,762,786,812,822,834,868,894,906,942,978,992,1002,1036,1038,1074,1086,1146,1148,1158,1182,1194,1204,1266,1316,1338,1362,1374,1398,1434,1446,1484
%N A275082 Primitive abundant numbers (having no abundant proper divisors) that have perfect proper divisors.
%C A275082 This sequence results from removing the terms of A071395 from A091191 (which are the two primitive abundant number sequences).
%C A275082 Each term of this sequence will be even (unless there exists an odd perfect number), be a multiple of a perfect number (A000396), and have just one perfect proper divisor.
%C A275082 If N is an even perfect number, then N = 2^(p-1)*M_p, where p is prime (A000043) and M_p = 2^p-1 is a Mersenne prime (A000668).  Abundant numbers of the form 2*N and q*N, where q is a prime number greater than or equal to M_p, will have no abundant proper divisors and only one perfect proper divisor (which is N).
%H A275082 Amiram Eldar, <a href="/A275082/b275082.txt">Table of n, a(n) for n = 1..10000</a>
%e A275082 a(5) = 56 = 2*28, since one of its proper divisors is a perfect number (28) and the rest are deficient numbers (1, 2, 4, 7, 8, 14).
%e A275082 a(13) = 196 = 7*28, since one of its proper divisors is a perfect number (28) and the rest are deficient numbers (1, 2, 4, 7, 14, 49, 98).
%t A275082 abdiv[n_] := (DivisorSigma[1, #] - 2#)& /@ Most@Divisors[n]; aQ[n_] := DivisorSigma[1, n] > 2n &&  AllTrue[(v = abdiv[n]), #<=0 &] && AnyTrue[v, #==0 &]; Select[Range[1500], aQ] (* _Amiram Eldar_, Jun 26 2019 *)
%Y A275082 Cf. A000043, A000396, A000668, A071395, subsequence of A091191.
%K A275082 nonn
%O A275082 1,1
%A A275082 _Timothy L. Tiffin_, Jul 15 2016