A067873 Numbers x satisfying x^2 - D*y^2 = 1 for more than one value of D distinct from x^2 - 1.

7, 9, 15, 17, 19, 23, 25, 26, 31, 33, 35, 37, 39, 41, 47, 49, 51, 53, 55, 57, 63, 65, 71, 73, 79, 80, 81, 82, 87, 89, 91, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 119, 121, 125, 127, 129, 135, 137, 143, 145, 149, 151, 153, 159, 161, 163, 167, 169, 170, 175, 177

Offset

1,1

Comments

The sequence contains A056220(n) for n>1. For a given x, solutions (x,y) to x^2 - D*y^2 = 1 do not all stand for the least nontrivial positive solution pair. For instance, x=244 solves the latter equation for D=15 and D=135 with corresponding y=63 and y=21. While (244,21) is indeed the fundamental solution to x^2 - 135*y^2 = 1, (244,63) comes only as the third one after the solution pairs (4,1) and (31,8) to x^2 - 15*y^2 = 1.

Sequence includes mostly odd entries in consecutive pairs. It seems reasonable to conjecture that an even entry e satisfies big omega(E) - omega(E) > 2, where E = e^2 - 1. (See A001222 and A001221.)

Links

Table of n, a(n) for n=1..61.

Prog

(PARI) { for(x=2, 10^3, f=core(x^2-1, 1)[2]; if(f>1 && !isprime(f), print1(x, ", ")) ) } \\ Max Alekseyev, Jul 25 2009

Crossrefs

Cf. A056220.

Sequence in context: A020939 A282758 A073457 * A217460 A347436 A047522

Adjacent sequences: A067870 A067871 A067872 * A067874 A067875 A067876

Keyword

nonn

Author

Lekraj Beedassy, Feb 25 2002

Extensions

Corrected and extended by Max Alekseyev, Jul 25 2009

Status

approved