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

12, 18, 30, 42, 56, 66, 78, 102, 114, 138, 174, 186, 196, 222, 246, 258, 282, 308, 318, 354, 364, 366, 402, 426, 438, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 654, 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

Offset: 1

Author

Timothy L. Tiffin, Jul 15 2016

Keywords

• nonn

Comments

This sequence results from removing the terms of A071395 from A091191 (which are the two primitive abundant number sequences).

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.

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).

Examples

			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).
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).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000043, A000396, A000668, A071395, subsequence of A091191.

Programs

Mathematica

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 *)