# [seqfan] Re: Anagrams of triangular numbers

Hans Havermann pxp at rogers.com
Sat Oct 17 20:26:34 CEST 2009

```zak seidov wrote:

> I calculated minimal T with exactly n anagrams, up to n=26,
> with yet not known T for n=15, 19-25.

Without supplying your minimal T's, here is my overview of what new
n's will show up for each (increasing) triangular-number digit-length
(*):

* 1   {1}
* 2   {}
* 3   {2}
* 4   {3}
* 5   {4}
* 6   {5, 6, 9}
* 7   {7, 8}
* 8   {10, 11, 12, 14, 16, 17}
* 9   {13, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 32}
*10   {28, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
48, 50, 51, 54, 55, 61, 71, 76, 77, 82}
*11   {47, 49, 52, 53, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68,
69, 70, 72, 74, 75, 78, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91,
94, 95, 96, 97, 98, 99, 100, 101, 102, 104, 107, 109, 114, 117, 119,
123, 133, 134, 145, 225}
*12   {73, 88, 92, 93, 103, 105, 106, 108, 110, 111, 112, 113, 115,
116, 118, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130, 131, 132,
135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 146, 147, 148, 149,
150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163,
164, 165, 166, 167, 169, 173, 174, 175, 176, 177, 178, 179, 180, 181,
183, 186, 187, 188, 189, 190, 192, 193, 195, 196, 198, 200, 201, 202,
203, 205, 207, 208, 209, 210, 212, 214, 215, 218, 219, 220, 222, 223,
226, 232, 233, 240, 243, 244, 246, 247, 250, 254, 256, 262, 269, 271,
308, 318, 339, 354, 385, 412}

One can see from this list that not only will nine-digit triangulars
supply your missing n=15, 19-25, they will also supply examples of n =
27, 29, 30, 32.

> Twenty six anagrams of triangular numbers:
>
> 102538360, 105306328, 106353820, 135062830, 165338020, 206583301,
> 238001653, 283160503, 300186253, 301658203, 303281506, 332085106,
> 350820316, 353128600, 368032015, 506813203, 586103203, 631528030,
> 802301653, 806031325, 806513203, 813032650, 821036503, 830301625,
> 832503610, 853320016
>
> Can be record broken?

The n = 412 maximum for 12-digit triangulars are: {101392475086,
101539782046, 102463891705, ... 976030128541, 981503476201,
985104730261}. ;)

```