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A076336 (Provable) Sierpiński numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite. 39
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 Sierpiński number. - T. D. Noe, Oct 31 2003

Sierpiński 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 Sierpiński in this way are non-Sierpiński. However, some numbers resist both proof and disproof. - David W. Wilson, Jan 17 2005

Sierpiński 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).

a(1) = 78557 is also the smallest odd n for which either n^p*2^k + 1 or n^p + 2^k is composite for every k > 0 and every prime p greater than 3. - Arkadiusz Wesolowski, Oct 12 2015

REFERENCES

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.

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.

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

G. Jaeschke, On the Smallest k Such that All k*2^N + 1 are Composite, Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 381-384.

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]

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

Seventeen or Bust, A Distributed Attack on the Sierpinski Problem

Eric Weisstein's World of Mathematics, Sierpiński Number of the Second Kind

CROSSREFS

Cf. A003261, A052333, A076335, A076337, A101036, A137715, A263169.

Sequence in context: A249084 A038826 A038815 * A244562 A123159 A184230

Adjacent sequences:  A076333 A076334 A076335 * A076337 A076338 A076339

KEYWORD

nonn,hard,nice,changed

AUTHOR

N. J. A. Sloane, Nov 07 2002

STATUS

approved

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Last modified February 9 02:39 EST 2016. Contains 268100 sequences.