A071395 Primitive abundant numbers (abundant numbers all of whose proper divisors are deficient numbers).

20, 70, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030, 4070, 4095, 4216, 4288

Offset: 1

Author

Joe McCauley (mccauley(AT)davesworld.net), Jun 12 2002

Keywords

• nonn

Comments

This is a subsequence of the primitive abundant number sequence A091191, since none of these numbers are a positive integer multiple of a perfect number (A000396). - Timothy L. Tiffin, Jul 15 2016

If the terms of this sequence are removed from A091191, then the resulting sequence will be A275082. - Timothy L. Tiffin, Jul 16 2016

Numbers n such that A294927(n) = 0 and A294937(n) = 1. - Antti Karttunen, Nov 14 2017

Examples

			20 is a term since 1, 2, 4, 5, and 10 (the proper divisors of 20) are all deficient numbers. - _Timothy L. Tiffin_, Jul 15 2016
		

References

• Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 46, also section B2, 1994.

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Primitive Abundant Number

Crossrefs

Cf. A006038, A000396, A005100, A005101, subsequence of A091191, A275082.

Cf. A294927, A294937.

Programs

Maple

abundance:= proc(n) option remember;  numtheory:-sigma(n)-2*n end proc:
select(n -> abundance(n) > 0 and andmap(t -> abundance(t) < 0, numtheory:-divisors(n) minus {n}), [$1..10000]); # Robert Israel, Nov 15 2017

Mathematica

Select[Range@ 5000, DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ #] == 1 &] (* Michael De Vlieger, Jul 16 2016 *)

PARI

isA071395(v) = {if (sigma(v) <= 2*v, return (0)); fordiv (v, d, if ((d != v) && (sigma(d) >= 2*d), return (0));); return (1);} \\ Michel Marcus, Mar 10 2013

Extensions

Offset corrected by Donovan Johnson, Aug 28 2011