A173330 - OEIS

A173330

First of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

1, 10, 1, 5, 1, 5, 46, 5, 70, 5, 9, 1, 106, 106, 126, 142, 146, 13, 9, 186, 1, 214, 13, 226, 1, 13, 9, 5, 17, 13, 306, 9, 5, 17, 366, 17, 378, 1, 406, 406, 17, 442, 21, 442, 5, 510, 21, 538, 13, 1, 570, 5, 17, 598, 25, 13, 25, 650, 1, 5, 694, 706, 9, 742, 25, 17, 786, 5, 25

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OFFSET

1,2

COMMENTS

A002972(n) = MIN(a(n), A002144(n) - a(n)).

REFERENCES

Harold Davenport, The Higher Arithmetic (Cambridge University Press 7th ed., 1999), ch. V.3, p. 122.

LINKS

Table of n, a(n) for n=1..69.

FORMULA

a(n) = (2k)! / 2(k!)^2 mod p, where p = 4*k+1 = A002144(n).

EXAMPLE

For n=7: A002144(7) = 53 = 4*13 + 1, a(7) = 26! / (2*(13!)^2) mod 53 = 46, and we have A002972(7) = MIN(46, 53 - 46) = 7;

For n=8: A002144(8) = 61 = 4*15 + 1, a(8) = 30! / (2*(15!)^2) mod 61 = 5, and we have A002972(8) = MIN(5, 61 - 5) = 5.

MATHEMATICA

p=Select[Prime[Range[142]], Mod[#, 4]==1 &]; Table[Mod[((Part[p, i]-1)/2)!/(2*((Part[p, i]-1)/4)!^2), Part[p, i]], {i, Length[p]}] (* Stefano Spezia, Dec 20 2025 *)

CROSSREFS

Cf. A173331, A001700, A010050, A000142, A005098.

Sequence in context: A082263 A337927 A131790 * A321097 A015810 A010183

Adjacent sequences: A173327 A173328 A173329 * A173331 A173332 A173333

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Feb 16 2010

STATUS

approved