[seqfan] Re: (Not)Possible values of sod of squares

David Wilson dwilson at gambitcomm.com
Wed Aug 26 21:33:46 CEST 2009

```Actually, your sod(n^2) will be {0,1,4,7} (mod 9).

It is easy that sod(n) == n (mod 9). So sod(n^2) == n^2 (mod 9), and
sod(n^2) must be a square (mod 9). This means that sod(n^2) is in {0, 1,
4, 7} (mod 9). I cannot immediately prove that all such numbers are
sod(n^2), but I highly suspect they are.

zak seidov wrote:
> Dear seqfans,
>
> The list of minimal m with sod(m^2)=n:
> {n,m}:
> {1,1},{4,2},{7,4},{9,3},{10,8},{13,7},{16,13},{18,24},{19,17},{22,43},{25,67},{27,63},{28,134},{31,83},{34,167},{36,264},{37,314},{40,313},{43,707},{45,1374},{46,836},{49,1667},{52,2236},{54,3114},{55,4472},{58,6833},{61,8167},{63,8937},{64,16667},{67,21886},{70,29614},{72,60663},{73,41833},{76,74833},{79,89437},{81,94863},{82,134164},{85,191833},{88,298327},{90,545793},{91,547613},{94,947617},{97,987917},{99,1989417},{100,1643167},{103,3143167},{106,3162083},{108,5477133},{109,9272917},{112,9893887},{115,19672313},{117,20736417},{118,24060133},{121,29983327},{124,44271886},{126,82395387},{127,60827617},{130,99477133},{133,197222917},{135,260191833},{136,197483417},{139,434738887},{142,529027313},{144,706399164},{145,1370363813},{148,994927133}.
> E.g., m(148) = 994927133 because m^2=994927133^2=989879999979599689
> and sod(m^2)=(9+8+9+8+7+9+9+9+9+9+7+9+5+9+9+6+8+9)=148.
>
> Not all n's<148 are present.
> Values of n which are multiples of 3^(2k+1) (odd powers of 3: 3,27,243) can not be sod of squares
> but besides them we have the list of "probable" non_sods_of_squares <148  :
>
> 2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83,86,89,92,95,98,101,104,107,110,113,116,119,122,125,128,131,134,137,140,143.
>
>  For n<100 the list of these non_sods_of_squares
> is apparently  full and correct while
> for larger values  the list needs rechecking.
> Thanks, Zak
>
>
>
>
>
>
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>
>

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