A374158 - OEIS

%I #38 Jul 02 2024 10:20:00

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

%N 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.

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

%C a(11) <= 3672178237.

%C a(12) = 6916.

%C a(13) = 33124.

%C a(14) = 29059303.

%C a(15) = 124852.

%C a(16) = 1983163.

%C a(18) = 48412.

%C a(20) = 18384457.

%C a(21) = 6117748.

%C a(22) = 1623076.

%C a(24) = 214396.

%C a(27) = 629356.

%C a(28) = 900838393.

%C a(31) = 79530724.

%C a(32) = 85276009.

%C a(37) = 274299844.

%C a(42) = 116237212.

%C a(60) = 73537828.

%C a(67) = 585930436.

%C From _Chai Wah Wu_, Jun 29-30 2024: (Start)

%C a(30) = 2372188.

%C a(36) = 1500772.

%C a(40) = 11957764.

%C a(45) = 30838444.

%C a(48) = 7932652.

%C a(54) = 19510036.

%C a(72) = 55528564.

%C (End)

%e    n | a(n)

%e -----+---------------------------

%e    1 |      4 = 2^2.

%e    2 |     91 = 7 * 13.

%e    3 |     28 = 2^2 * 7.

%e    4 |    196 = 2^2 * 7^2.

%e    5 |  31213 = 7^4 * 13.

%e    6 |    364 = 2^2 * 7 * 13.

%e    7 |   9604 = 2^2 * 7^4.

%e    8 |  53599 = 7 * 13 * 19 * 31.

%e    9 |   2548 = 2^2 * 7^2 * 13.

%e   10 | 470596 = 2^2 * 7^6.

%o (Python)

%o from itertools import count

%o from sympy.abc import x,y

%o from sympy.solvers.diophantine.diophantine import diop_quadratic

%o 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

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

%Y Cf. A002476, A092573.

%K nonn,more

%O 0,2

%A _Seiichi Manyama_, Jun 29 2024