OFFSET
1,3
COMMENTS
The length of row n is 1 for n = 1 and n = 2, and for n >= 3 it is 2^{r1 + r4} with the number r1 and r4 of distinct primes congruent to 1 and 4 modulo 5, respectively, in the prime number factorization of N(n). E.g., n = 29, N = 209 = 11*19, has r1 = 1 and r4 = 1, with four solutions.
For N(1) = 1 every integer solves this Diophantine equation, and the representative solution is 0.
For N(2) = 5 there is only one representative solution, namely 0.
For n >= 3 the solutions come in a nonnegative power of 2 pairs, each of the type (x1, x2) with x2 = N - x1.
See the link in A089270 to the W. Lang paper, section 3, and Table 7.
EXAMPLE
The irregular triangle T(n, k) begins (pairs (x, N - x) in brackets):
n, N \ k 1 2 3 4 ...
----------------------------------
1, 1: 0
2, 5: 0
3, 11: (4 7)
4, 19: (9 10)
5, 29: (11 18)
6, 31: (6 25)
7, 41: (13 28)
8, 55: (15 40)
9, 59: (8 51)
10, 61: (26 35)
11, 71: (17 54)
12, 79: (20 59)
13, 89: (19 70)
14, 95: (10 85)
15, 101: (45 56)
16, 109: (21 88)
17, 121: (48 73)
18, 131: (23 108)
19, 139: (12 127)
20, 145: (40 105)
....
29, 209: (29 180) (48 161)
...
41, 319: (18 301) (40 279)
...
43, 341: (37 304) (161 180)
...
59, 451: (95 356) (136 315)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Wolfdieter Lang, Jul 08 2019
STATUS
approved