A374095 -id:A374095 - cp's OEIS frontend

A374094 a(n) is the smallest nonnegative integer k where there are exactly n solutions to x^2 + x*y + y^2 = k with 0 < x < y.

Original entry on oeis.org

0, 7, 91, 637, 1729, 31213, 12103, 405769, 53599, 157339, 593047

Offset: 0

Seiichi Manyama, Jun 28 2024

a(n) is the smallest nonnegative k such that A374092(k) = n.

a(11) > 10^8. - Robert Israel, Jun 28 2024

Cf. A093195, A374095.

Cf. A374090, A374092.

N:= 10^6:
V:= Array(0..N):
for x from 1 to floor(sqrt(N/3)) do
  for y from x+1 do
     v:= x^2 + x*y + y^2;
     if v > N then break fi;
     V[v]:= V[v]+1;
od od:
W:= Array(0..10);
for i from 1 to N while count < 11 do
  v:= V[i];
  if W[v] = 0 then W[v]:= i; count:= count+1 fi
od:
0, seq(W[i],i=1..10); # Robert Israel, Jun 28 2024

from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374094(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x*(x+y)+y**2-m) if 0Chai Wah Wu, Jun 28 2024

a(n) <= 13 * 7^(n-1).

A374091 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + 3xy + y^2 = k.

Original entry on oeis.org

0, 5, 11, 605, 209, 73205, 2299, 8857805, 6061, 218405, 278179

Offset: 0

Seiichi Manyama, Jun 28 2024

a(n) is the smallest nonnegative k such that A374089(k) = n.
a(n) is of the form 5*k^2 if and only if n is odd.
a(12) = 66671.
a(14) = 5285401.
a(15) = 26427005.
a(16) = 187891.
a(18) = 1266749.
a(20) = 8067191.
a(21) = 3197667605. - Chai Wah Wu, Jun 29 2024
a(24) = 2066801.
a(26) = 36735721.
a(27) = 183678605. - Chai Wah Wu, Jun 28 2024

Cf. A328151, A374090.

Cf. A374089, A374095.

a(2*n) = A374095(n).

a(n) <= 5*11^(n-1) for all n >= 1. - Jason Yuen, Jun 29 2024

A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3xy + y^2 with 0 <= x <= y.

Original entry on oeis.org

2, 0, 121, 2299, 6061, 43681, 66671, 33659659, 187891, 1266749, 8067191, 639533521, 2066801

Offset: 0

Seiichi Manyama, Jul 02 2024

a(n) is the smallest nonnegative k such that A374276(k) = n.
a(14) = 36735721.
a(15) = 153276629.
a(16) = 7703531.
a(18) = 39269219.
a(20) = 250082921.
a(24) = 84738841.
a(32) = 454508329.

Cf. A000446, A198799.

Cf. A031363, A374091, A374095, A374276.

cp's OEIS Frontend

A374094 a(n) is the smallest nonnegative integer k where there are exactly n solutions to x^2 + x*y + y^2 = k with 0 < x < y.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Python

Formula

A374091 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + 3xy + y^2 = k.

Views

Author

Keywords

Comments

Crossrefs

Formula

A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3xy + y^2 with 0 <= x <= y.

Views

Author

Keywords

Comments

Crossrefs

cp's OEIS Frontend

A374094 a(n) is the smallest nonnegative integer k where there are exactly n solutions to x^2 + x*y + y^2 = k with 0 < x < y.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Python

Formula

A374091 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + 3*x*y + y^2 = k.

Views

Author

Keywords

Comments

Crossrefs

Formula

A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3*x*y + y^2 with 0 <= x <= y.

Views

Author

Keywords

Comments

Crossrefs

A374091 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + 3xy + y^2 = k.

A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3xy + y^2 with 0 <= x <= y.