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%I #12 Oct 08 2018 18:10:35
%S 1,5,8,12,36,54,56,87,95,160,212,328,342,356,427,531,660,672,843,852,
%T 858,909,910,940,992,1002,1012,1162,1222,1245,1353,1417,1435,1495,
%U 1509,1547,1757,1837,1909,1927,1998,2072,2274,2793,2983,3051,3212,3219,3515,3548,3870
%N Numbers n such that the sum of divisors of n is a hexagonal number.
%C This is to A006532 (numbers n such that sum of divisors is a square) as A000326 (Pentagonal numbers) is to A000290 (squares); and as A180929 (Numbers n such that the sum of divisors of n is a pentagonal number) is to A000326 (Pentagonal numbers); and as A045746 (Numbers n such that the sum of divisors of n is a triangular number) is to A000217 (triangular numbers), and as A000384 (Hexagonal numbers) is to A000290 (squares).
%C 54, 56, 87, and 95 are the smallest four numbers whose sum of divisors is the same hexagonal number (120).
%H Charles R Greathouse IV, <a href="/A180930/b180930.txt">Table of n, a(n) for n = 1..10000</a>
%F A000203(a(n)) is in A000384. sigma(a(n) = k*(2*k-1) for some nonnegative integer k. Sum of divisors of a(n) is a hexagonal number. sigma_1(a(n)) is a hexagonal number.
%e a(1) = 1 because the sum of divisors of 1 is the hexagonal number 1.
%e a(2) = 5 because the sum of divisors of 5 is the hexagonal number 6.
%e a(3) = 8 because the sum of divisors of 8 is the hexagonal number 15.
%e a(4) = 12 because the sum of divisors of 12 is the hexagonal number 28.
%p isA000384 := proc(n) if not issqr(8*n+1) then false; else sqrt(8*n+1)+1 ; (% mod 4) = 0 ; end if; end proc:
%p for n from 1 to 4000 do if isA000384(numtheory[sigma](n)) then printf("%d,",n) ; end if; end do: # _R. J. Mathar_, Sep 26 2010
%t hnos=Table[n (2n-1),{n,500}]; okQ[n_]:=Module[{ds=DivisorSigma[1,n]},MemberQ[hnos,ds]] Select[Range[5000],okQ] (* _Harvey P. Dale_, Sep 26 2010 *)
%Y Cf. A000203, A000384, A006532, A045746, A180929.
%K nonn
%O 1,2
%A _Jonathan Vos Post_, Sep 26 2010
%E Corrected and extended by several authors, Sep 27 2010
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