A091191 Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor.

12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114, 138, 174, 186, 196, 222, 246, 258, 272, 282, 304, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 550, 572, 582, 606, 618, 642, 644, 650, 654, 678, 748, 762, 786, 812, 822

Offset: 1

Author

Reinhard Zumkeller, Dec 27 2003

Keywords

• nonn

Comments

A080224(a(n)) = 1.

This is a supersequence of the primitive abundant number sequence A071395, since many of these numbers will be positive integer multiples of the perfect numbers (A000396). - Timothy L. Tiffin, Jul 15 2016

If the terms of A071395 are removed from this sequence, then the resulting sequence will be A275082. - Timothy L. Tiffin, Jul 16 2016

Examples

			12 is a term since 1, 2, 3, 4, and 6 (the proper divisors of 12) are either deficient or perfect numbers, and thus not abundant. - _Timothy L. Tiffin_, Jul 15 2016
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• P. Erdős, On the density of the abundant numbers, J. London Math. Soc. 9 (1934), pp. 278-282.
• Eric Weisstein's World of Mathematics, Abundant Number

Crossrefs

Cf. A006038 (odd terms), A005101 (abundant numbers), A091192.

Cf. A027751, A071395 (subsequence), supersequence of A275082.

Cf. A294930 (characteristic function), A294890.

Programs

Haskell

a091191 n = a091191_list !! (n-1)
a091191_list = filter f [1..] where
   f x = sum pdivs > x && all (<= 0) (map (\d -> a000203 d - 2 * d) pdivs)
         where pdivs = a027751_row x
-- Reinhard Zumkeller, Jan 31 2014

Maple

isA005101 := proc(n) is(numtheory[sigma](n) > 2*n ); end proc:
isA091191 := proc(n) local d; if isA005101(n) then for d in numtheory[divisors](n) minus {1,n} do if isA005101(d) then return false; end if; end do: return true; else false; end if; end proc:
for n from 1 to 200 do if isA091191(n) then printf("%d\n",n) ; end if;end do: # R. J. Mathar, Mar 28 2011

Mathematica

t = {}; n = 1; While[Length[t] < 100, n++; If[DivisorSigma[1, n] > 2*n && Intersection[t, Divisors[n]] == {}, AppendTo[t, n]]]; t (* T. D. Noe, Mar 28 2011 *)
Select[Range@ 840, DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &] (* Michael De Vlieger, Jul 16 2016 *)

PARI

is(n)=sumdiv(n,d,sigma(d,-1)>2)==1 \\ Charles R Greathouse IV, Dec 05 2012

Formula

Erdős shows that a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 05 2012