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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.

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A344232 All positive integers k properly represented by the positive definite binary quadratic form 2*X^2 + 2*X*Y + 3*Y^2 = k, in increasing order.

Original entry on oeis.org

2, 3, 7, 10, 15, 18, 23, 27, 35, 42, 43, 47, 58, 63, 67, 82, 83, 87, 90, 98, 103, 107, 115, 122, 123, 127, 135, 138, 147, 162, 163, 167, 178, 183, 202, 203, 207, 210, 215, 218, 223, 227, 235, 243, 258, 263, 267, 282, 283, 287, 290, 298, 303, 307, 315, 322, 327, 335, 343, 347, 362, 367, 378, 383, 387
Offset: 1

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Author

Wolfdieter Lang, Jun 10 2021

Keywords

Comments

This is one of the bisections of sequence A343238. The other sequence is A344231.
This is a proper subsequence of A029718.
The primes in this sequence are given in A106865.
See A344231 for more details.
The reduced form [2, 2, 3] represents the proper (determinant +1) equivalence class of one of the two genera (genus II) of discriminant -20. The multiplicative generic characters for discriminant Disc = -20 have values Jacobi(a(n)|5) = -1 and Jacobi(-1|a(n)) = -1, for odd a(n) not divisible by 5. See Buell, p. 52.
The product of any two odd a(n), not divisible by 5, is congruent to {1,5} (mod 8). See Buell, 4), p. 51.
For this genus II of Disc = -20 the positive integers represented are given by 2^a*5^b*Product_{j=1..PI} (pI_j)^(eI(j))*Product_{k=1..PII}(pII_k)^(eII(k)), with a and b from {0, 1}, but if PI = PII = 0 (empty products are 1) then (a, b) = (1, 0) or (1, 1), giving a(1) = 2 or a(4) = 10. The odd primes pI_j are from A033205 and the odd primes pII_j from the odd primes of A106865. The exponents of the second product satisfy: if a = 1 then PII >= 0, and if PII >=1 then Sum_{k=1..PII} eII(j) is even. If a = 0 then PII >= 1 and this sum is odd.
The neighboring numbers k (twins) begin: [42, 43], [82, 83], [122, 123] [162, 163], [202, 203], [282, 283], ...
For the solutions (X, Y) of F2 = [2, 2, 3] properly representing k = a(n) see A344234.

References

  • D. A. Buell, Binary Quadratic Forms, Springer, 1989.
  • A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, Sammlung Göschen Band 5131, Walter de Gruyter, 1973.

Crossrefs

Cf. A029718, A033205, A106865, A343238, A343239, A343240, A344231, A344234.

A344233 Irregular triangle read by rows: row n gives the pairs of proper solutions (X, Y), with gcd(X, Y) = 1 and X >= 0, of the Diophantine equation X^2 + 5*Y^2 = A344231(n), for n >= 1.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, -1, 2, 1, 2, -1, 3, 1, 3, -1, 1, 2, 1, -2, 4, 1, 4, -1, 3, 2, 3, -2, 5, 1, 5, -1, 6, 1, 6, -1, 5, 2, 5, -2, 1, 3, 1, -3, 2, 3, 2, -3, 7, 1, 7, -1, 4, 3, 3, -3, 8, 1, 8, -1, 7, 2, 7, -2, 5, 3, 5, -3, 1, 4, 1, -4, 9, 1, 9, -1, 3, 4, 3, -4, 7, 3, 7, -3
Offset: 1

Views

Author

Wolfdieter Lang, May 17 2021

Keywords

Comments

The length of row n is r(n) = 2*A343240(b(n)), if A344231(n) = A343238(b(n)), for n >= 1. This sequence begins 2*(1, 1, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, ...).
The number of proper solutions (X, Y), with X > 0, is 1 for n = 1. X = 0 only for n = 2, and for n >= 2 only one half of the solutions are listed, namely those with X >= 0. There are also the solutions with (-X, -Y). Thus the total number of solutions for n >= 2 is actually r(n) given above.
For n >= 2 each distinct odd primes from {1, 3, 7, 9} (mod 20), i.e., from
A139513, that divides A344231(n) contributes a factor 2 to the number of solutions listed here. See A343238 and the corresponding A343240.

Examples

			The irregular triangle T(n, m) begins (A(n) = A344231(n)):
n   A(n) \ m  1  2   3  4   5  6   7  8 ...
1,   1:       1  0
2,   5:       0  1
3,   6:       1  1   1 -1
4,   9:       2  1   2 -1
5,  14:       3  1   3 -1
6,  21:       1  2   1 -2   4  1   4 -1
7,  29:       3  2   3 -2
8,  30:       5  1   5 -1
9,  41:       6  1   6 -1
10, 45:       5  2   5 -2
11, 46:       1  3   1 -3
12, 49:       2  3   2 -3
13, 54:       7  1   7 -1
14, 61:       4  3   3 -3
15, 69:       8  1   8 -1   7  2   7 -2
16, 70:       5  3   5 -3
17, 81:       1  4   1 -4
18, 86:       9  1   9 -1
19, 89:       3  4   3 -4
20, 94:       7  3   7 -3
...
n = 2: Prime 5 is not a member of A139513, therefore only 1 solution appears here (see the remark above on the solution (0, -1)).
n = 4: Prime 3 a member A139513. Thus there are 2^1 = 2 solutions are listed. The solution (3, 0) does not appear; it is not proper.
n = 6: 21 = A344231(6) = A343238(12) = 3*7. hence A343240(12) = 2^2 = 4 and there are 4 pairs of proper solutions with X >= 0.

Crossrefs

Cf. A343238, A343240, A344231, A344234.

Formula

T(n, m) gives for m = 2^j-1, the nonnegative X(n, j) solution, and for m = 2*j the Y(n, j) solution of T(n, 2*j-1)^2 + 5*T(n, 2*j)^2 = A344231(n), for j = 1 ..r(n), for n >= 1. For n = 2 (X(2) = 0) the solution (0, -1) is not listed.
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