cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jun 28 2024

Keywords

Comments

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)

Crossrefs

Cf. A002476, A328151, A374091.
Cf. A003136, A374088, A374094.

Programs

  • Python
    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

Formula

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

Extensions

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

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.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jun 28 2024

Keywords

Comments

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

Crossrefs

Cf. A093195, A374094.
Cf. A045468, A374091, A374093.

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.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jul 02 2024

Keywords

Comments

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.

Crossrefs

Cf. A000446, A198799.
Cf. A031363, A374091, A374095, A374276.
Showing 1-3 of 3 results.

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