A173331 Second of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

2, 2, 13, 2, 31, 4, 2, 55, 8, 81, 4, 91, 99, 105, 133, 10, 6, 2, 10, 181, 183, 227, 8, 237, 16, 10, 14, 265, 2, 301, 303, 16, 18, 8, 355, 379, 6, 381, 389, 14, 421, 429, 453, 451, 487, 20, 531, 543, 20, 24, 585, 24, 18, 16, 637, 631, 655, 12, 651, 675, 22, 731, 26, 741, 757

Offset: 1

Author

Reinhard Zumkeller, Feb 16 2010

Keywords

• nonn

Comments

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

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

Examples

			n=7: A002144(7) = 53 = 4*13 + 1,
a(7) = A173330(7) * 26! mod 53 = 7*403291461126605635584000000 mod 53 = 2,
A002973(7) = MIN(2, 53 - 2) / 2 = 1;
n=8: A002144(8) = 61 = 4*15 + 1,
a(8) = A173330(8) * 30! mod 61 = 5*265252859812191058636308480000000 mod 61 = 55,
A002973(8) = MIN(55, 61 - 55) / 2 = 3.
		

References

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

Crossrefs

Cf. A123072, A000142.

Formula

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