A160556 Positive integers b for which the Diophantine equation f = (b^(2n) - b^n + 8n^2 - 2) / (2n * (2n + 1)) has at least ten solutions for n <= 10000

2, 8, 14, 17, 26, 29, 32, 38, 41, 47, 50, 59, 62, 64, 65, 68, 74, 77, 83, 89, 95, 98, 101, 104, 110, 119, 122, 128, 131, 134, 137, 140, 143, 149, 152, 155, 161, 164, 167, 173, 179, 182, 185, 188, 194, 197, 200, 206, 209, 212, 215, 218, 221, 224, 227, 230, 233

Offset

1,1

Comments

For these equations (not exclusively), the sequences of 2n + 1 are dominated by primes.

When b = 2, there are 105 solutions with n less than 10000, and in this case, the sequence of n is also dominated by primes: only five of these are composite. The average difference between successive composite terms is near the magnitude of n. No composite values of 2n + 1 have been found. n and 2n + 1 account for roughly 3% of primes less than 20 billion. For other bases, n is almost always composite, and 2n + 1 is almost always prime.

The next most productive values of b less than 1000 are 509 (41 solutions) and 824 (40 solutions).

Bases that produce a greater or equal number of solutions than smaller bases, except 2, often have ones digit 4 or 9. Values of n associated with composite 2n + 1 are often divisible by 5.

Links

Table of n, a(n) for n=1..57.

Crossrefs

Cf. A158034, A158035, A158036

Cf. A160557

Sequence in context: A194278 A050619 A056715 * A105610 A324691 A117104

Adjacent sequences: A160553 A160554 A160555 * A160557 A160558 A160559

Keyword

easy,nonn

Author

Reikku Kulon, May 19 2009

Status

approved