A091192 -id:A091192 - cp's OEIS frontend

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

Reinhard Zumkeller, Dec 27 2003

nonn

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

			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

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

Cf. A006038 (odd terms), A005101 (abundant numbers), A091192.
Cf. A027751, A071395 (subsequence), supersequence of A275082.
Cf. A294930 (characteristic function), A294890.

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

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

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 *)

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

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

A080224 Number of abundant divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 1, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Feb 07 2003

nonn

Number of divisors d of n with sigma(d)>2*d (sigma = A000203)

a(n)>0 iff n is abundant: a(A005101(n))>0, a(A000396(n))=0 and a(A005100(n))=0; a(A091191(n))=1; a(A091192(n))>1; a(A091193(n))=n and a(m)<>n for m < A091193(n). - Reinhard Zumkeller, Dec 27 2003

			Divisors of n=24: {1,2,3,4,6,8,12,24}, two of them are abundant: 12=A005101(1) and 24=A005101(4), therefore a(24)=2.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Abundant Number.

Cf. A000005, A000203, A005101, A080225, A080226, A187795, A294890, A294929, A294930, A294937.

Cf. also A294904.

A080224 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if numtheory[sigma](d) > 2*d then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A080224(n),n=1..80) ; # R. J. Mathar, Feb 22 2021

Table[Count[Divisors[n],?(DivisorSigma[1,#]>2#&)],{n,110}] (* _Harvey P. Dale, Jun 14 2013 *)

a(n) = sumdiv(n, d, sigma(d)>2*d)  \\ Michel Marcus, Mar 09 2013

a(n) + A080225(n) + A080226(n) = A000005(n).
From Antti Karttunen, Nov 14 2017: (Start)
a(n) = Sum_{d|n} A294937(d).
a(n) = A294929(n) + A294937(n).
a(n) = 1 iff A294930(n) = 1.
(End)

A337688 Numbers divisible by 2 or more primitive nondeficient numbers.

Original entry on oeis.org

60, 84, 120, 140, 168, 180, 210, 240, 252, 264, 280, 300, 312, 336, 360, 420, 440, 480, 504, 520, 528, 540, 560, 588, 600, 616, 624, 630, 660, 672, 700, 720, 728, 756, 780, 792, 816, 840, 880, 900, 912, 924, 936, 960, 980, 1008, 1020, 1040, 1050, 1056, 1080, 1092

Offset: 1

David A. Corneth, Antti Karttunen and Peter Munn, Sep 20 2020

A006039 lists the primitive nondeficient numbers.

The first odd term is 4725.

			Table of n, a(n) and the relevant divisors starts:
   n   a(n)   divisors in A006039
   1     60   6, 20;
   2     84   6, 28;
   3    120   6, 20;
   4    140   20, 28, 70;
   5    168   6, 28;
   6    180   6, 20;
   7    210   6, 70;
   8    240   6, 20;
   9    252   6, 28;
  10    264   6, 88;
  11    280   20, 28, 70;
  12    300   6, 20;
  13    312   6, 104;
  14    336   6, 28;
  15    360   6, 20;
  16    420   6, 20, 28, 70;

David A. Corneth, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program

A006039 is used to define this sequence.
Subsequence of: A091192.
A337689 is a subsequence.
A337690 is used in a formula defining this sequence.

See Corneth link \\ David A. Corneth, Oct 07 2020

Numbers k such that A337690(k) >= 2.

cp's OEIS Frontend

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

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A080224 Number of abundant divisors of n.

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A337688 Numbers divisible by 2 or more primitive nondeficient numbers.

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