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A180930
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Numbers n such that the sum of divisors of n is a hexagonal number.
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1
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1, 5, 8, 12, 36, 54, 56, 87, 95, 160, 212, 328, 342, 356, 427, 531, 660, 672, 843, 852, 858, 909, 910, 940, 992, 1002, 1012, 1162, 1222, 1245, 1353, 1417, 1435, 1495, 1509, 1547, 1757, 1837, 1909, 1927, 1998, 2072, 2274, 2793, 2983, 3051, 3212, 3219, 3515, 3548, 3870
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OFFSET
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1,2
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COMMENTS
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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).
54, 56, 87, and 95 are the smallest four numbers whose sum of divisors is the same hexagonal number (120).
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LINKS
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FORMULA
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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.
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EXAMPLE
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a(1) = 1 because the sum of divisors of 1 is the hexagonal number 1.
a(2) = 5 because the sum of divisors of 5 is the hexagonal number 6.
a(3) = 8 because the sum of divisors of 8 is the hexagonal number 15.
a(4) = 12 because the sum of divisors of 12 is the hexagonal number 28.
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MAPLE
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isA000384 := proc(n) if not issqr(8*n+1) then false; else sqrt(8*n+1)+1 ; (% mod 4) = 0 ; end if; end proc:
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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Corrected and extended by several authors, Sep 27 2010
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STATUS
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approved
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