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

1,1

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).

Links

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

Example

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).

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 *)

Crossrefs

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

Sequence in context: A344198 A112054 A225576 * A256753 A167597 A138636

Adjacent sequences: A275079 A275080 A275081 * A275083 A275084 A275085

Keyword

nonn

Author

Timothy L. Tiffin, Jul 15 2016

Status

approved