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A076336 (Provable) Sierpinski numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite. 27
78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, 3083723, 3098059, 3555593, 3608251 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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. - T. D. Noe, 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).

REFERENCES

Dan Ismailescu and Peter Seho Park, On Pairwise Intersections of the Fibonacci, Sierpinski, and Riesel Sequences, Journal of Integer Sequences, 16 (2013), #13.9.8.

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

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

W. Sierpinski, Sur un probleme concernant les nombres k * 2^n + 1, Elem. Math., 15 (1960), pp. 63-74.

LINKS

T. D. Noe and Arkadiusz Wesolowski, Table of n, a(n) for n = 1..15000 (T. D. Noe supplied 13394 terms which came from McLean. a(1064), a(7053), and a(13397)-a(15000) from Arkadiusz Wesolowski.)

Chris Caldwell, Riesel number

Chris Caldwell, Sierpinski number

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

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

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 Number of the Second Kind

CROSSREFS

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

Sequence in context: A249084 A038826 A038815 * A244562 A123159 A184230

Adjacent sequences:  A076333 A076334 A076335 * A076337 A076338 A076339

KEYWORD

nonn,hard,nice

AUTHOR

N. J. A. Sloane, Nov 07 2002

STATUS

approved

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Last modified October 21 06:29 EDT 2014. Contains 248375 sequences.