A374094 -id:A374094 - cp's OEIS frontend

A374090 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + x*y + y^2 = k.

0, 3, 7, 147, 91, 7203, 637, 352947, 1729, 24843, 31213, 847425747, 12103, 41523861603, 405769, 1217307, 53599, 99698791708803, 157339, 4885240793731347, 593047, 59648043

Offset: 0

Seiichi Manyama, Jun 28 2024

a(n) is the smallest nonnegative k such that A374088(k) = n.
From Chai Wah Wu, Jun 28 2024: (Start)
If x <> y and x^2 + x*y + y^2 = a(n), then (x, y) and (y, x) both count as solutions. Therefore if a(n) exists, then a(n) is of the form 3*m^2 if and only if n is odd. This also implies that a(2*n) = A374094(n).
a(25)  = 205724883.
a(27)  = 8968323.
a(33)  = 143214951243.
a(35)  = 10080519267.
a(45)  = 439447827.
a(49)  = 1703607756123.
a(63)  = 21532943523.
a(75)  = 74266682763.
a(81)  = 8618558403.
a(135) = 422309361747.
(End)
From David A. Corneth, Jun 29 2024: (Start)
a(19) <= 3*7^18.
a(22) <= 3672178237.
a(24)  = 375193.
a(26)  = 2989441 <= 179936733613.
a(28)  = 29059303.
a(30)  = 7709611.
a(32)  = 1983163.
a(34) <= 432028097404813.
a(36)  = 4877509.
Conjecture: Let q_i be the i-th prime of the form 3*k + 1 and let m = Prod_{j=1, t} b_j, a factorization of m into factors > 1.
Let f(m) = Prod_{j = 1..t} q_i^(b_(t+1-j)-1).
Then for even n we have a(n) = min(f(n), f(n+1))
and for odd n we have a(n) = 3*f(n).
Example for n = 22 we might factor 22 = 11*2. The first two primes of the form 3*k + 1 are 7 and 13. So we would have a(22) = min(7^10*13, 7^22).
a(14) = min(f(14), f(15)) = min(7^6 * 13, 7^4 * 13^2) = 405769. (End)

Cf. A002476, A328151, A374091.

Cf. A003136, A374088, A374094.

from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374090(n): return next(m for m in (3*k**2 if n&1 else k for k in count(0)) if sum(1 for d in diop_quadratic(x*(x+y)+y**2-m) if d[0]>0 and d[1]>0) == n) # Chai Wah Wu, Jun 28 2024

a(2*n) = A374094(n).

a(11), a(13) from Chai Wah Wu, Jun 28 2024
a(17) from Bert Dobbelaere, Jun 28 2024
a(19) from Bert Dobbelaere, Jun 30 2024

A374092 Number of solutions to n = x^2 + x*y + y^2 with 0 < x < y.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jun 28 2024

nonn

a(n) = 0 if n == 2 (mod 4). - Robert Israel, Jun 28 2024

Cf. A025441, A374093, A374094.

N:= 200: # for a(0) .. a(N)
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:
convert(V,list); # Robert Israel, Jun 28 2024

A374095 a(n) is the smallest nonnegative integer k where exactly n solutions to x^2 + 3xy + y^2 = k with 0 < x < y.

Original entry on oeis.org

0, 11, 209, 2299, 6061, 278179, 66671, 5285401, 187891, 1266749, 8067191

Offset: 0

Seiichi Manyama, Jun 28 2024

a(n) is the smallest nonnegative k such that A374093(k) = n.

Cf. A093195, A374094.

Cf. A045468, A374091, A374093.

cp's OEIS Frontend

A374090 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + x*y + y^2 = k.

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A374092 Number of solutions to n = x^2 + x*y + y^2 with 0 < x < y.

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A374095 a(n) is the smallest nonnegative integer k where exactly n solutions to x^2 + 3xy + y^2 = k with 0 < x < y.

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cp's OEIS Frontend

A374090 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + x*y + y^2 = k.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

Formula

Extensions

A374092 Number of solutions to n = x^2 + x*y + y^2 with 0 < x < y.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

A374095 a(n) is the smallest nonnegative integer k where exactly n solutions to x^2 + 3*x*y + y^2 = k with 0 < x < y.

Views

Author

Keywords

Comments

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A374095 a(n) is the smallest nonnegative integer k where exactly n solutions to x^2 + 3xy + y^2 = k with 0 < x < y.