OFFSET
1,1
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
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
FORMULA
Erdős shows that a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 05 2012
EXAMPLE
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
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 *)
PROG
(PARI) is(n)=sumdiv(n, d, sigma(d, -1)>2)==1 \\ Charles R Greathouse IV, Dec 05 2012
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Dec 27 2003
STATUS
approved