A374160 -id:A374160 - cp's OEIS frontend

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

Seiichi Manyama, Jun 29 2024

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)

			   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.

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

Cf. A002476, A092573.

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

A374159 a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 7*y^2 = k.

Original entry on oeis.org

0, 8, 32, 128, 352, 704, 1408, 2816, 5632, 11264, 16192, 45056, 32384, 123904, 64768, 178112, 129536, 2883584, 259072, 1982464, 469568, 712448, 1036288, 184549376, 939136, 21551552, 4145152, 2849792, 1878272

Offset: 0

Seiichi Manyama, Jun 29 2024

a(n) is the smallest nonnegative k such that A216511(k) = n.
Conjecture: All terms are multiple of a(1) = 8.
a(30) = 5165248.
a(31) = 16386304.
a(32) = 3756544.
a(33) = 11399168.
a(34) = 66322432.
a(35) = 86206208.
a(36) = 7513088.

Cf. A328151, A343105, A374158, A374160, A374161.

Cf. A216511.

from itertools import count
from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374159(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x**2+7*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

Seiichi Manyama, Jun 29 2024

Cf. A328151, A343105, A374158, A374159, A374160.

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

A374017 Number of solutions to x^2 + 11*y^2 = n in positive integers x and y.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2

Offset: 0

Seiichi Manyama, Jun 29 2024

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A092573, A216279, A216511.

Cf. A374160.

a(n) = if(n==0, 0, sum(k=1, sqrtint((n-1)\11), issquare(n-11*k^2)));

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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.

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A374159 a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 7*y^2 = k.

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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.

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A374017 Number of solutions to x^2 + 11*y^2 = n in positive integers x and y.

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