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

%I A091191 #37 Feb 16 2025 08:32:52
%S A091191 12,18,20,30,42,56,66,70,78,88,102,104,114,138,174,186,196,222,246,
%T A091191 258,272,282,304,308,318,354,364,366,368,402,426,438,464,474,476,498,
%U A091191 532,534,550,572,582,606,618,642,644,650,654,678,748,762,786,812,822
%N A091191 Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor.
%C A091191 A080224(a(n)) = 1.
%C A091191 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
%C A091191 If the terms of A071395 are removed from this sequence, then the resulting sequence will be A275082. - _Timothy L. Tiffin_, Jul 16 2016
%H A091191 T. D. Noe, <a href="/A091191/b091191.txt">Table of n, a(n) for n = 1..10000</a>
%H A091191 P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1934-04.pdf">On the density of the abundant numbers</a>, J. London Math. Soc. 9 (1934), pp. 278-282.
%H A091191 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbundantNumber.html">Abundant Number</a>
%F A091191 Erdős shows that a(n) >> n log^2 n. - _Charles R Greathouse IV_, Dec 05 2012
%e A091191 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
%p A091191 isA005101 := proc(n) is(numtheory[sigma](n) > 2*n ); end proc:
%p A091191 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:
%p A091191 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 A091191 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 *)
%t A091191 Select[Range@ 840, DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &] (* _Michael De Vlieger_, Jul 16 2016 *)
%o A091191 (PARI) is(n)=sumdiv(n,d,sigma(d,-1)>2)==1 \\ _Charles R Greathouse IV_, Dec 05 2012
%o A091191 (Haskell)
%o A091191 a091191 n = a091191_list !! (n-1)
%o A091191 a091191_list = filter f [1..] where
%o A091191    f x = sum pdivs > x && all (<= 0) (map (\d -> a000203 d - 2 * d) pdivs)
%o A091191          where pdivs = a027751_row x
%o A091191 -- _Reinhard Zumkeller_, Jan 31 2014
%Y A091191 Cf. A006038 (odd terms), A005101 (abundant numbers), A091192.
%Y A091191 Cf. A027751, A071395 (subsequence), supersequence of A275082.
%Y A091191 Cf. A294930 (characteristic function), A294890.
%K A091191 nonn
%O A091191 1,1
%A A091191 _Reinhard Zumkeller_, Dec 27 2003