1,1

It is only a conjecture that this sequence is complete up to 3000000 - there may be missing terms.

It is conjectured that 78557 is the smallest Sierpinski number. These numbers are from Joseph McLean, who has computed 13535 Sierpinski numbers less than 2*10^9. - T. D. Noe (noe(AT)sspectra.com), Oct 31 2003

Sierpinski numbers are proved by exhibiting a periodic sequence p of prime divisors with p(k) | n*2^k+1 and disproved by finding prime n*2^k+1. It is conjectured that numbers that cannot be proved Sierpinski in this way are non-Sierpinski. However, some numbers resist both proof and disproof. - David W. Wilson, Jan 17 2005.

Sierpinski showed that this sequence is infinite.

There are 4 related sequences that arise in this context:

S1: Numbers n such that n*2^k + 1 is composite for all k (this sequence)

S2: Odd numbers n such that 2^k + n is composite for all k (apparently it is conjectured that S1 and S2 are the same sequence)

S3: Numbers n such that n*2^k + 1 is prime for all k (empty)

S4: Numbers n such that 2^k + n is prime for all k (empty)

The following argument, kindly provided by Michael Reid, shows that S3 and S4 are empty:

If p is a prime divisor of n + 1, then for k = p - 1, the term (either n*2^k + 1 or 2^k + n ) is a multiple of p (and also > p, so not prime).

P. Ribenboim, The Book of Prime Number Records, 2nd. ed., 1989, p. 282.

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 420.

T. D. Noe, Table of n, a(n) for n=0..13535 (from McLean)

Chris Caldwell, Riesel Numbers

Chris Caldwell, Sierpinski Numbers

Yves Gallot, A search for some small Brier numbers, 2000.

J. McLean, Searching for large Sierpinski numbers [Broken link?]

J. McLean, Searching for large Sierpinski numbers [Cached copy]

J. McLean, Brier Numbers [Broken link?]

J. McLean, Brier Numbers [Cached copy]

C. Rivera, Brier numbers

Payam Samidoost, Dual Sierpinski problem search page [Broken link?]

Payam Samidoost, Dual Sierpinski problem search page [Cached copy]

Payam Samidoost, 4847 [Broken link?]

Payam Samidoost, 4847 [Cached copy]

Seventeen or Bust, A Distributed Attack on the Sierpinski problem

Eric Weisstein's World of Mathematics, Sierpinski numbers

Cf. A076337, A076335, A003261, A052333, A101036.

Sequence in context: A017587 A038826 A038815 this_sequence A123159 A157661 A159713

Adjacent sequences: A076333 A076334 A076335 this_sequence A076337 A076338 A076339

nonn,hard,nice

N. J. A. Sloane (njas(AT)research.att.com), Nov 07 2002