

A180929


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


2



1, 6, 11, 104, 116, 129, 218, 363, 408, 440, 481, 534, 566, 568, 590, 638, 646, 684, 718, 807, 895, 979, 999, 1003, 1007, 1137, 1251, 1282, 1557, 1935, 2197, 2367, 2571, 2582, 2808, 2855, 3132, 3283, 3336, 3578, 3737, 3891, 3946, 3980, 4172, 4484, 4886, 5158
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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 A045746 Numbers n such that the sum of divisors of n is a triangular number, is to A000217 triangular numbers. 6 and 11 are the first two numbers whose sum of divisors is the same pentagonal number. 104 and 116 are the second two numbers whose sum of divisors is the same pentagonal number. 363 and 481 are the third two numbers whose sum of divisors is the same pentagonal number. 408, 440, 534, 568, 590, 638, 646, 718, 807, 895, 979, 1003, and 1007 are the first thirteen numbers whose sum of divisors is the same pentagonal number.
The subsequence of values which are themselves pentagonal numbers begins a(1) = 1, a(15) = 590, 395403426.


LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000


FORMULA

A000203(a(n)) is in A000326. sigma(a(n) = k*(3*k1)/2 for some nonnegative integer k. Sum of divisors of a(n) is a pentagonal number. sigma_1(a(n)) is a pentagonal number.


EXAMPLE

a(1) = 1 because the sum of divisors of 1 is the pentagonal number 1.
a(2) = 6 because the sum of divisors of 6 is the pentagonal number 12.
a(3) = 11 because the sum of divisors of 11 is the pentagonal number 12.
a(4) = 104 because the sum of divisors of 104 is the pentagonal number 210.
a(5) = 116 because the sum of divisors of 116 is the pentagonal number 210.
a(6) = 129 because the sum of divisors of 129 is the pentagonal number 176.
a(7) = 218 because the sum of divisors of 218 is the pentagonal number 330.


MAPLE

# Contribution from R. J. Mathar, Sep 26 2010: (Start)
isA000326 := proc(n) if not issqr(24*n+1) then false; else sqrt(24*n+1)+1 ; (% mod 6) = 0 ; end if; end proc:
for n from 1 to 5000 do if isA000326(numtheory[sigma](n)) then printf("%d, ", n) ; end if; end do: # (End)


MATHEMATICA

pnos=Table[(n (3n1))/2, {n, 500}]; okQ[n_]:=Module[{ds=DivisorSigma[1, n]}, MemberQ[pnos, ds]] Select[Range[5000], okQ] (* Harvey P. Dale, Sep 26 2010 *)
Select[Range[5200], IntegerQ[(Sqrt[1+24DivisorSigma[1, #]]+1)/6]&] (* Harvey P. Dale, Jun 14 2013 *)


CROSSREFS

Cf. A000203, A000217, A000290, A000326, A006532, A045746.
Sequence in context: A217934 A012419 A012663 * A212592 A351991 A318858
Adjacent sequences: A180926 A180927 A180928 * A180930 A180931 A180932


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Sep 25 2010


EXTENSIONS

Corrected and extended by R. J. Mathar and Harvey P. Dale, Sep 26 2010


STATUS

approved



