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A347832 Irregular triangle T, read by rows, giving the solutions x for x*(x + 1) == -4 (mod A347831 (n)), for x from {0, 1, 2, ..., A347831(n)-1}, for n >= 1. 2
0, 0, 1, 1, 0, 3, 2, 1, 4, 3, 4, 2, 7, 4, 7, 7, 3, 12, 5, 11, 8, 10, 7, 12, 6, 16, 4, 19, 7, 22, 13, 17, 12, 19, 5, 11, 22, 28, 8, 10, 27, 29, 12, 27, 6, 16, 29, 39, 9, 37, 19, 28, 22, 28, 20, 32, 10, 46, 7, 52, 15, 45, 13, 17, 44, 48, 19, 44, 11, 28, 39, 56, 16, 52, 8, 27, 48, 67, 35, 43, 12, 67, 31, 51 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

T(n, k) gives the solutions x from {0, 1, ..., A347831(n) - 1} of the congruence (x + 1)*x + 4 == 0 (mod A347831(n)), for n >= 1. No other positive modulus has a solution.

The length of row n of the triangle is A347833(n).

The present congruence 2*T(x) + 4 == 0 (mod k), for k >= 1, with the triangular numbers T(n) = A000217(n), is equivalent to the congruence s^2 + 15 == 0 (mod 4*k) where s = 2*x + 1. Each of these two congruences has a solution for k >= 1 if and only if k is prepresented by some positive definite binary quadratic form of discriminant disc = -15. See e.g., Buell Proposition 41, p. 50, or Scholz-Schoeneberg Satz 74, p. 105.

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.

LINKS

Table of n, a(n) for n=1..84.

EXAMPLE

The irregular triangle T with A(n) = A347831(n) begins:

n A(n) \ k 1 2 3 4 ...

1, 1: 0

2, 2: 0 1

3, 3: 1

4, 4: 0 3

5, 5: 2

6, 6: 1 4

7, 8: 3 4

8, 10: 2 7

9, 12: 4 7

10, 15: 7

11, 16: 3 12

12, 17: 5 11

13, 19: 8 10

14, 20: 7 12

15, 23: 6 16

16, 24: 4 19

17, 30: 7 22

18, 31: 6 35

19, 32: 12 19

20, 34: 5 11 22 28

...

PROG

(PARI) isok(m) = {my(f=factor(m)); for (k=1, #f~, my(p=f[k, 1]); if ((p==3) || (p==5), if (f[k, 2] > 1, return (0)), if (kronecker(p, 15) != 1, return(0))); ); return (1); } \\ A347831

tabf(nn) = {for (n=1, nn, if (isok(n), for (x=0, n-1, if (Mod(x*(x+1), n) == -4, print1(x, ", ")); ); ); ); } \\ Michel Marcus, Oct 23 2021

CROSSREFS

Cf. A000217, A347831, A347833.

Sequence in context: A190698 A283183 A327467 * A077427 A107641 A299352

Adjacent sequences: A347829 A347830 A347831 * A347833 A347834 A347835

KEYWORD

nonn,tabf,easy

AUTHOR

Wolfdieter Lang, Sep 15 2021

STATUS

approved

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Last modified December 4 16:41 EST 2022. Contains 358563 sequences. (Running on oeis4.)