A387129 Second numbers B = a(n) of two numbers (A, B) such that the sums 2*A^2 + B^2 = p == 3 mod 8, where p = A007520(n) and A = A387128(n).

1, 3, 1, 5, 3, 7, 9, 3, 9, 11, 1, 9, 7, 15, 3, 11, 17, 13, 3, 19, 9, 21, 15, 21, 7, 19, 23, 15, 11, 3, 13, 25, 9, 21, 23, 17, 25, 19, 27, 29, 1, 5, 15, 27, 21, 13, 33, 31, 21, 17, 33, 3, 15, 29, 27, 33, 27, 1, 5, 21, 39, 37, 39, 11, 39, 13, 33, 41, 29, 17, 27, 33, 43, 15, 3, 27, 23, 9

Offset: 1

Author

Vladimir Pletser, Aug 17 2025

Keywords

• nonn
• new

Comments

Prime numbers p congruent to 3 mod 8 can be written as the sum of twice the square of an integer A and of the square of another integer B, i.e., 2*A^2 + B^2 = p, where A = A387128(n), B = a(n) (this sequence), and p = A007520(n) == 3 mod 8.

This representation is unique, i.e., for a given n, there are no other integer values of A(n) and B(n) such that p(n) = 2 * A(n)^2 + B(n)^2 where p(n) = A007520(n), the 3 mod 8 prime numbers.

For all n, A = A387128(n) and B = a(n) are always odd.

Terms are ordered according to increasing order of A007520(n).

Examples

			1 belongs to the sequence as 2 * 1^2 + 1^2 = 3.
21 belongs to the sequence as 2 * 5^2 + 21^2 = 491.
		

References

• Cartier P. "An Introduction to Zeta Functions", Chap 1.2, in eds. M. Waldschmidt, P. Moussa, J.M., Luck, C. Itzykson “From Number Theory to Physics”, Springer-Verlag, Berlin, pp. 22-41, 1960.
• Conway J.H. and Guy R.K. "The Book of Numbers", Chap. 5, Springer-Verlag, New York, pp. 127-149, 1996.
• Hardy, G. H. and Wright, E. M. "Primes in k(i)" and "The Fundamental Theorem of Arithmetic in k(i)." 12.7 and 12.8 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 183-187, 1979.
• Sierpinski W. "Elementary Theory of Numbers", Chap. 13.3 and 13.4, ed. A Schinzel, North Holland, Amsterdam, pp. 459-462, 1988.

Links

• Vladimir Pletser, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007520, A387128.

Formula

2 * A387128(n)^2 + a(n)^2 = A007520(n).