A096822 Smallest primes of form p = 2^x-(2n-1) where x=A096502(n), the least exponent providing this kind of prime.

3, 5, 3, 549755813881, 7, 5, 3, 17, 47, 13, 11, 41, 7, 5, 3, 97, 31, 29, 2011, 89, 23, 536870869, 19, 17, 79, 13, 11, 73, 7, 5, 3, 193, 191, 61, 59, 953, 439, 53, 179, 433, 47, 173, 43, 41, 167, 37, 163, 929, 31, 29, 67108763, 409, 23, 149, 19, 17, 911, 13, 11, 137

Offset

1,1

Comments

If 2n-1 is a provable Riesel number (A101036), then there exists a finite set of primes P(2n-1) such that every 2^x-(2n-1) > 0 is divisible by p(x) in P(2n-1). If some 2^x-(2n-1) = p(x), then a(n) = p(x). Otherwise, p(x) is a proper divisor of 2^x-(2n-1), which must be composite, and no a(n) exists.

For example, if n = 254602, then 2n-1 = 509203 is a provable Riesel number. Every 2^x-509203 > 0 is divisible by prime p(x) in P(509203) = {3,5,7,13,17,241}. 2^x-509203 > 0 implies x >= 19 implies 2^x-509203 > 241 >= p(x), so p(x) is a proper divisor and every 2^x-509203 is composite. Hence a(254602) does not exist.

Links

T. D. Noe, Table of n, a(n) for n = 1..935

Example

a(1) = 3 is the first Mersenne prime;
a(64) = 2^47 - 127 = 140737488355201, where 47 = A096502(64), 127 = 2*64 - 1.

Mathematica

f[n_]:=Module[{lst={}, exp=Ceiling[Log[2, 1+n]]}, While[!PrimeQ[2^exp-n], exp++]; AppendTo[lst, 2^exp-n]]; Flatten[f/@Range[1, 1001, 2]] (* Ivan N. Ianakiev, Mar 08 2016 *)

Crossrefs

Cf. A096502.

Sequence in context: A195418 A366857 A065974 * A195383 A303069 A086567

Adjacent sequences: A096819 A096820 A096821 * A096823 A096824 A096825

Keyword

nonn

Author

Labos Elemer, Jul 13 2004

Status

approved