A173331 - OEIS

A173331

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

2, 11, 13, 2, 31, 4, 51, 55, 65, 81, 4, 91, 10, 8, 4, 139, 151, 2, 10, 12, 183, 2, 8, 4, 16, 10, 14, 265, 2, 301, 14, 16, 18, 8, 18, 379, 391, 381, 20, 407, 421, 20, 453, 10, 487, 501, 531, 14, 20, 24, 8, 24, 18, 601, 637, 631, 655, 661, 651, 675, 687, 2, 26, 20, 757, 751, 26, 781, 14

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OFFSET

1,1

REFERENCES

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

Harold Davenport, The Higher Arithmetic, Cambridge University Press, 8th ed., 2008, p. 109.

LINKS

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

FORMULA

a(n) = A173330(n)*A010050(A005098(n)) mod A002144(n).

A002973(n) = MIN(a(n), A002144(n) - a(n)) / 2.

a(n) = A173330(n) * (2*k)! (mod p), where p = 4*k+1 = A002144(n) [corrected by Stefano Spezia, Dec 20 2025]

EXAMPLE

For n=7: A002144(7) = 53 = 4*13 + 1, a(7) = A173330(7) * 26! mod 53 = 46*403291461126605635584000000 mod 53 = 51, and we have A002973(7) = MIN(2, 53 - 2) / 2 = 1;

For n=8: A002144(8) = 61 = 4*15 + 1, a(8) = A173330(8) * 30! mod 61 = 5*265252859812191058636308480000000 mod 61 = 55, and we have A002973(8) = MIN(55, 61 - 55) / 2 = 3.

MATHEMATICA

p=Select[Prime[Range[150]], Mod[#, 4]==1 &]; x=Table[Mod[((Part[p, i]-1)/2)!/(2*((Part[p, i]-1)/4)!^2), Part[p, i]], {i, Length[p]}]; Table[Mod[((Part[p, i]-1)/2)!Part[x, i], Part[p, i]], {i, Length[p]}] (* Stefano Spezia, Dec 20 2025 *)

CROSSREFS

Cf. A000142, A002144, A002973, A005098, A010050, A123072, A173330.

Sequence in context: A037091 A289675 A127303 * A378841 A286153 A068972

Adjacent sequences: A173328 A173329 A173330 * A173332 A173333 A173334

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Feb 16 2010

EXTENSIONS

Corrected by Stefano Spezia, Dec 20 2025

STATUS

approved