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

Jonathan Post jvospost3 at gmail.com
Sun Sep 26 04:07:37 CEST 2010

```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
>
>
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>

```

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