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

Max Alekseyev maxale at gmail.com
Sun Sep 26 06:48:14 CEST 2010

```Here are the terms below 10^5:

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, 5174, 5235,
5282, 5296, 5811, 5892, 6297, 6445, 6995, 7163, 7167, 7172, 7343,
7453, 7538, 7787, 7839, 7961, 8027, 8159, 8201, 8275, 9182, 9898,
10595, 11479, 11626, 12047, 12127, 12205, 12779, 13529, 13644, 14381,
15380, 16971, 17094, 17230, 17378, 17451, 17803, 17876, 17904, 18210,
19994, 20686, 20691, 20838, 21018, 21146, 22056, 22331, 22785, 23254,
23271, 23835, 23931, 24346, 24480, 24608, 24915, 25209, 25226, 25234,
25291, 25522, 25855, 26085, 26103, 26106, 26262, 26436, 27229, 27345,
27400, 27922, 27975, 28077, 28083, 28166, 28342, 28348, 28531, 28702,
28707, 28746, 29182, 29212, 29291, 29785, 30017, 30063, 30816, 30915,
30957, 31115, 31529, 31885, 31937, 32007, 32163, 32196, 32436, 33104,
33385, 33523, 34779, 34922, 34937, 34956, 35185, 35335, 35485, 36182,
36547, 37177, 37280, 37809, 37958, 38203, 38297, 38533, 38810, 39114,
39997, 40533, 40806, 41397, 41546, 41929, 42377, 42587, 42817, 43097,
43137, 43279, 43357, 45995, 46281, 46487, 46582, 46991, 47536, 47668,
47861, 48293, 48607, 49396, 49536, 49723, 50562, 51523, 51725, 51847,
52956, 53282, 53871, 54221, 54683, 54731, 55658, 56023, 56356, 56361,
57396, 57482, 57518, 57856, 58881, 58966, 59076, 59448, 59855, 60176,
61227, 63708, 64407, 64707, 65748, 65977, 66097, 67302, 69584, 70101,
70779, 70780, 73088, 73125, 73705, 74032, 74275, 74610, 75978, 77746,
80097, 80540, 80636, 80797, 81607, 81859, 82121, 83308, 85194, 86761,
87005, 89181, 90838, 91706, 92014, 92727, 93301, 94025, 94126, 94356,
95077, 95372, 96234, 96882, 96941, 97238, 97353, 98100, 98134, 99081,
99118

Regards,
Max

On Sat, Sep 25, 2010 at 11:22 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> 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/
>>>
>>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

```