[seqfan] Re: n-th number whose sum of divisors is a n-gonal number

[Jonathan Post]

For sigma(a(n)) pentagonal, I think we have:

%I A180929
%S A180929 1,6,11,104,116,129,218,363,408,440,481,534,566,568,590,638,646,684,718,
%T A180929 807,895,979,999,1003,1007,1137,1251,1282,1557,2197
%N A180929 Numbers n such that the sum of divisors of n is a pentagonal number.
%C A180929 This is to A006532 (numbers n such that sum of divisors is
a square) as A000326 Pentagonal numbers is to A000290 squares, and as
A180927 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.
%F A180929 A000203(a(n)) is in A000326. sigma(a(n) = k*(3*k-1)/2 for
some nonnegative integer k. Sum of divisors of a(n) is a pentagonal
number. sigma_1(a(n)) is a pentagonal number.
%e A180929 a(1) = 1 because the sum of divisors of 1 is the pentagonal number 1.
%e A180929 a(2) = 6 because the sum of divisors of 6 is the pentagonal
number 12.
%e A180929 a(3) = 11 because the sum of divisors of 11 is the
pentagonal number 12.
%e A180929 a(4) = 104 because the sum of divisors of 104 is the
pentagonal number 210.
%e A180929 a(5) = 116 because the sum of divisors of 116 is the
pentagonal number 210.
%e A180929 a(6) = 129 because the sum of divisors of 129 is the
pentagonal number 176.
%e A180929 a(7) = 218 because the sum of divisors of 218 is the
pentagonal number 330.
%Y A180929 Cf. A000203, A000217, A000290, A000326, A006532, A180927.
%K A180929 easy,more,nonn
%O A180929 1,2
%A A180929 Jonathan Vos Post (jvospost3(AT)gmail.com), Sep 25 2010



On Sat, Sep 25, 2010 at 7:07 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> Dear Douglas McNeil,
>
> You're right as to my missing values.
> sigma(12) = 28
> sigma(87) = 120
> sigma(95) = 120
> so 87 and 95 are the second smallest pair of numbers whose sum of
> divisors is the same triangular number.
> Hence you have clearly corrected and extended A180927 Numbers n such
> that the sum of divisors of n is a triangular number.
>
> You may be the Geologist, but I have feet of clay.
>
> Would you like to enter that by form for njas et al?
>
> For the A[n,n] I can't be sure, polygonality (as you put it) being
> known by formula, but I'm not familiar with sage.
>
> Thank you!
>
> On Sat, Sep 25, 2010 at 5:47 PM, Douglas McNeil <mcneil at hku.hk> wrote:
>>> %S A180927 1,2,5,8,22,36,45,54,56,98
>>> %N A180927 Numbers n such that the sum of divisors of n is a triangular number.
>>
>> I think 12, 87, and 95 are missing.  I find
>>
>> sage: a[:100]
>> [1, 2, 5, 8, 12, 22, 36, 45, 54, 56, 87, 95, 98, 104, 116, 152, 160,
>> 200, 212, 258, 328, 342, 356, 393, 427, 441, 473, 492, 531, 572, 582,
>> 588, 660, 668, 672, 726, 740, 800, 843, 852, 858, 879, 908, 909, 910,
>> 940, 962, 992, 1002, 1012, 1067, 1079, 1162, 1197, 1222, 1245, 1272,
>> 1353, 1417, 1435, 1469, 1495, 1496, 1509, 1517, 1547, 1614, 1664,
>> 1757, 1790, 1837, 1909, 1927, 1944, 1957, 1958, 1998, 2006, 2014,
>> 2036, 2048, 2072, 2124, 2192, 2274, 2427, 2465, 2502, 2702, 2793,
>> 2895, 2959, 2983, 3043, 3051, 3078, 3103, 3114, 3127, 3212]
>>
>> As for the diagonal A[n,n], I get a sequence beginning (n >= 2)
>>
>> [2, 5, 66, 116, 54, 225, 108, 777, 880, 869, 253, 1870, 4068, 1658,
>> 699, 5050, 3327, 4770, 2577, 2992, 5219, 22810, 992, 7372, 22595,
>> 9938, 2482, 22982, 13796, 14094, 5534, 10766, 31588, 31242, 3590,
>> 36888, 29643, 33365, 3997, 88962, 43632, 5512, 21235, 8390, 47587,
>> 137689, 4342, 58507, 95341, 36384, 11745, 247156, 76745, 24849, 20109,
>> 32399, 104950, 174508, 10763, 49947, 218314, 118477, 15899, 464947,
>> 96772, 63388, 27559, 64368, 114806, 235416, 20547, 141331, 533106,
>> 111677, 29733, 474448, 120092, 268235, 46167, 75322, 296387, 328999,
>> 40364, 111414, 477569, 303574, 22553, 568861, 141606, 95932, 109527,
>> 270938, 382609, 772856, 18459, 543470, 701623, 229947, 52022]
>>
>> but it'd be easy to have made an error testing polygonality or an
>> off-by-one error or any of the usual mistakes.
>>
>>
>> Doug
>>
>> --
>> Department of Earth Sciences
>> University of Hong Kong
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>