A181471 a(n) = number of numbers of the form k^2-1 having n-th prime as largest prime divisor.

1, 4, 8, 16, 20, 34, 47, 72, 95, 126, 168, 208, 262, 343, 433, 507, 634, 799, 976, 1146, 1439, 1698, 2082, 2371, 2734

Offset: 1

Author

Artur Jasinski, Oct 21-22 2010

Keywords

• nonn
• hard
• more

Comments

Theorem: zero does not occur in this sequence. Proof: (p-1)^2-1=(p-2)p. This means that p is greatest prime divisor of (p-1)^2-1 for every p.

An effective abc conjecture (c < rad(abc)^2) would imply that a(24)-a(33) are (2371, 2734, 3360, 4022, 4637, 5575, 6424, 7268, 8351, 9661). - Lucas A. Brown, Oct 01 2022

Links

• Florian Luca and Filip Najman, On the largest prime factor of x^2-1, arXiv:1005.1533 [math.NT], 2010.
• Florian Luca and Filip Najman, On the largest prime factor of x^2-1, Mathematics of Computation 80 (2011), 429-435. (Paper has errata that was posted on the MOC website.)
• Wikipedia, Størmer's theorem.

Crossrefs

Row lengths of A223701.

Cf. A039915, A175607.

Extensions

Wrong terms a(24)-a(25) removed by Lucas A. Brown, Oct 01 2022

a(24)-a(25) from David A. Corneth, Oct 01 2022