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.
Original entry on oeis.org
0, 4, 91, 28, 196, 31213, 364, 9604, 53599, 2548, 470596
Offset: 0
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.
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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
A374160
a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 11*y^2 = k.
Original entry on oeis.org
0, 12, 60, 180, 540, 1620, 2700, 8100, 12420, 20700, 37260, 1180980, 62100, 476100, 335340, 186300, 310500, 1822500, 558900, 53144100, 931500, 1676700, 4284900, 324860625, 1925100, 4657500, 244462860, 12854700, 8383500
Offset: 0
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from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374160(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x**2+11*y**2-m) if d[0]>0 and d[1]>0)==n) # Chai Wah Wu, Jun 30 2024
A374161
a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 19*y^2 = k.
Original entry on oeis.org
0, 20, 140, 700, 1540, 17500, 7700, 122500, 26180, 53900, 192500, 7035875, 130900, 592900, 4812500, 1347500, 602140, 150062500, 916300
Offset: 0
-
from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374161(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x**2+19*y**2-m) if d[0]>0 and d[1]>0)==n) # Chai Wah Wu, Jun 30 2024
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