A341678 Irregular triangle read by rows: row n consists of all numbers x such that x^2 + y^2 = A006278(n), with 0 < x < y.

1, 1, 4, 4, 9, 12, 23, 2, 19, 46, 67, 74, 86, 109, 122, 64, 103, 167, 191, 236, 281, 292, 359, 449, 512, 568, 601, 607, 664, 673, 743, 59, 132, 531, 581, 627, 876, 1008, 1284, 1588, 1659, 1723, 2092, 2136, 2317, 2373, 2736, 2757, 2803, 3072, 3164, 3333, 3469, 3704, 3821, 4028, 4077, 4136, 4371, 4596, 4668, 4712, 4851

Offset: 1

Author

Richard Peterson, Feb 17 2021

Keywords

• nonn
• tabf

Comments

The n-th row of the triangle is of length 2^(n-1), since a product of n distinct primes congruent to 1 (mod 4) has 2^(n-1) solutions to being the sum of two squares.

Examples

			Triangle starts:
1,
1, 4,
4, 9, 12, 23,
2, 19, 46, 67, 74, 86, 109, 122,
64, 103, 167, 191, 236, 281, 292, 359, 449, 512, 568, 601, 607, 664, 673, 743,
...
In the second row, calculations are as follows. 5*13 is the product of the first two primes congruent to 1 (mod 4), and 65 = 1^2 + 8^2 = 4^2 + 7^2, so the second row is 1, 4.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1023 (rows 1..10)
• Eric Weisstein's World of Mathematics, Sum of Squares Function

Crossrefs

Cf. A006278, A230710.

Cf. A236381 (1st column).

Programs

PARI

row(n) = {my(t=1, q=3, v=vector(2^n/2)); for(k=1, n, until(q%4==1, q=nextprime(q+1)); t*=q); q=0; for(k=1, #v, until(issquare(t-q^2), q++); v[k]=q); v; } \\ Jinyuan Wang, Mar 03 2021