A374158 a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 3*y^2 = k.

0, 4, 91, 28, 196, 31213, 364, 9604, 53599, 2548, 470596

Offset: 0

Author

Seiichi Manyama, Jun 29 2024

Keywords

• nonn
• more

Comments

a(n) is the smallest nonnegative k such that A092573(k) = n.

a(11) <= 3672178237.

a(12) = 6916.

a(13) = 33124.

a(14) = 29059303.

a(15) = 124852.

a(16) = 1983163.

a(18) = 48412.

a(20) = 18384457.

a(21) = 6117748.

a(22) = 1623076.

a(24) = 214396.

a(27) = 629356.

a(28) = 900838393.

a(31) = 79530724.

a(32) = 85276009.

a(37) = 274299844.

a(42) = 116237212.

a(60) = 73537828.

a(67) = 585930436.

From Chai Wah Wu, Jun 29-30 2024: (Start)

a(30) = 2372188.

a(36) = 1500772.

a(40) = 11957764.

a(45) = 30838444.

a(48) = 7932652.

a(54) = 19510036.

a(72) = 55528564.

(End)

Examples

			   n | a(n)
-----+---------------------------
   1 |      4 = 2^2.
   2 |     91 = 7 * 13.
   3 |     28 = 2^2 * 7.
   4 |    196 = 2^2 * 7^2.
   5 |  31213 = 7^4 * 13.
   6 |    364 = 2^2 * 7 * 13.
   7 |   9604 = 2^2 * 7^4.
   8 |  53599 = 7 * 13 * 19 * 31.
   9 |   2548 = 2^2 * 7^2 * 13.
  10 | 470596 = 2^2 * 7^6.
		

Crossrefs

Cf. A328151, A343105, A374159, A374160, A374161.

Cf. A002476, A092573.

Programs

Python

from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374158(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x**2+3*y**2-m) if d[0]>0 and d[1]>0)==n) # Chai Wah Wu, Jun 29 2024