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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.

%I A374158 #38 Jul 02 2024 10:20:00
%S A374158 0,4,91,28,196,31213,364,9604,53599,2548,470596
%N 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.
%C A374158 a(n) is the smallest nonnegative k such that A092573(k) = n.
%C A374158 a(11) <= 3672178237.
%C A374158 a(12) = 6916.
%C A374158 a(13) = 33124.
%C A374158 a(14) = 29059303.
%C A374158 a(15) = 124852.
%C A374158 a(16) = 1983163.
%C A374158 a(18) = 48412.
%C A374158 a(20) = 18384457.
%C A374158 a(21) = 6117748.
%C A374158 a(22) = 1623076.
%C A374158 a(24) = 214396.
%C A374158 a(27) = 629356.
%C A374158 a(28) = 900838393.
%C A374158 a(31) = 79530724.
%C A374158 a(32) = 85276009.
%C A374158 a(37) = 274299844.
%C A374158 a(42) = 116237212.
%C A374158 a(60) = 73537828.
%C A374158 a(67) = 585930436.
%C A374158 From _Chai Wah Wu_, Jun 29-30 2024: (Start)
%C A374158 a(30) = 2372188.
%C A374158 a(36) = 1500772.
%C A374158 a(40) = 11957764.
%C A374158 a(45) = 30838444.
%C A374158 a(48) = 7932652.
%C A374158 a(54) = 19510036.
%C A374158 a(72) = 55528564.
%C A374158 (End)
%e A374158    n | a(n)
%e A374158 -----+---------------------------
%e A374158    1 |      4 = 2^2.
%e A374158    2 |     91 = 7 * 13.
%e A374158    3 |     28 = 2^2 * 7.
%e A374158    4 |    196 = 2^2 * 7^2.
%e A374158    5 |  31213 = 7^4 * 13.
%e A374158    6 |    364 = 2^2 * 7 * 13.
%e A374158    7 |   9604 = 2^2 * 7^4.
%e A374158    8 |  53599 = 7 * 13 * 19 * 31.
%e A374158    9 |   2548 = 2^2 * 7^2 * 13.
%e A374158   10 | 470596 = 2^2 * 7^6.
%o A374158 (Python)
%o A374158 from itertools import count
%o A374158 from sympy.abc import x,y
%o A374158 from sympy.solvers.diophantine.diophantine import diop_quadratic
%o A374158 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 A374158 Cf. A328151, A343105, A374159, A374160, A374161.
%Y A374158 Cf. A002476, A092573.
%K A374158 nonn,more
%O A374158 0,2
%A A374158 _Seiichi Manyama_, Jun 29 2024