

A180930


Numbers n such that the sum of divisors of n is a hexagonal number.


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

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).


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000


FORMULA

A000203(a(n)) is in A000384. sigma(a(n) = k*(2*k1) for some nonnegative integer k. Sum of divisors of a(n) is a hexagonal number. sigma_1(a(n)) is a hexagonal number.


EXAMPLE

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.


MAPLE

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


MATHEMATICA

hnos=Table[n (2n1), {n, 500}]; okQ[n_]:=Module[{ds=DivisorSigma[1, n]}, MemberQ[hnos, ds]] Select[Range[5000], okQ] (* Harvey P. Dale, Sep 26 2010 *)


CROSSREFS

Cf. A000203, A000384, A006532, A045746, A180929.
Sequence in context: A260966 A338547 A124434 * A185729 A164128 A259912
Adjacent sequences: A180927 A180928 A180929 * A180931 A180932 A180933


KEYWORD

nonn


AUTHOR

Jonathan Vos Post, Sep 26 2010


EXTENSIONS

Corrected and extended by several authors, Sep 27 2010


STATUS

approved



