cp's OEIS Frontend

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

%I A071395 #31 Feb 16 2025 08:32:46
%S A071395 20,70,88,104,272,304,368,464,550,572,650,748,836,945,1184,1312,1376,
%T A071395 1430,1504,1575,1696,1870,1888,1952,2002,2090,2205,2210,2470,2530,
%U A071395 2584,2990,3128,3190,3230,3410,3465,3496,3770,3944,4030,4070,4095,4216,4288
%N A071395 Primitive abundant numbers (abundant numbers all of whose proper divisors are deficient numbers).
%C A071395 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
%C A071395 If the terms of this sequence are removed from A091191, then the resulting sequence will be A275082. - _Timothy L. Tiffin_, Jul 16 2016
%C A071395 Numbers n such that A294927(n) = 0 and A294937(n) = 1. - _Antti Karttunen_, Nov 14 2017
%D A071395 Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 46, also section B2, 1994.
%H A071395 Donovan Johnson, <a href="/A071395/b071395.txt">Table of n, a(n) for n = 1..10000</a>
%H A071395 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimitiveAbundantNumber.html">Primitive Abundant Number</a>
%e A071395 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
%p A071395 abundance:= proc(n) option remember;  numtheory:-sigma(n)-2*n end proc:
%p A071395 select(n -> abundance(n) > 0 and andmap(t -> abundance(t) < 0, numtheory:-divisors(n) minus {n}), [$1..10000]); # _Robert Israel_, Nov 15 2017
%t A071395 Select[Range@ 5000, DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ #] == 1 &] (* _Michael De Vlieger_, Jul 16 2016 *)
%o A071395 (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
%Y A071395 Cf. A006038, A000396, A005100, A005101, subsequence of A091191, A275082.
%Y A071395 Cf. A294927, A294937.
%K A071395 nonn
%O A071395 1,1
%A A071395 Joe McCauley (mccauley(AT)davesworld.net), Jun 12 2002
%E A071395 Offset corrected by _Donovan Johnson_, Aug 28 2011