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A076336
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(Provable) Sierpiński numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite.
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61
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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
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OFFSET
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1,1
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COMMENTS
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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, due to Michael Reid, attempts to show 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). [However, David McAfferty points that for the case S3, this argument fails if p is of the form 2^m-1. So it may only be a conjecture that the set S3 is empty. - N. J. A. Sloane, Jun 27 2021]
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
n = 4008735125781478102999926000625 = (A213353(1))^4 is in this sequence but is thought not to satisfy the conjecture mentioned by David W. Wilson above. For this multiplier, all n*2^(4m + 2) + 1 are composite by an Aurifeuillean factorization. Only the remaining cases, n*2^k + 1 where k is not 2 modulo 4, are covered by a finite set of primes (namely {3, 17, 97, 241, 257, 673}). See Izotov link for details (although with another prime set). - Jeppe Stig Nielsen, Apr 14 2018
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REFERENCES
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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.
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LINKS
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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.
Anatoly S. Izotov, A Note on Sierpinski Numbers, Fibonacci Quarterly (1995), pp. 206-207.
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. 73-74.
Seventeen or Bust, A Distributed Attack on the Sierpinski Problem
Jeremiah T. Southwick, Two Inquiries Related to the Digits of Prime Numbers, Ph. D. Dissertation, University of South Carolina (2020).
Eric Weisstein's World of Mathematics, Sierpiński Number of the Second Kind
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CROSSREFS
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Cf. A003261, A052333, A076335, A076337, A101036, A137715, A263169, A305473, A306151.
Sequence in context: A249084 A038826 A038815 * A244562 A123159 A184230
Adjacent sequences: A076333 A076334 A076335 * A076337 A076338 A076339
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KEYWORD
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nonn,hard,nice
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AUTHOR
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N. J. A. Sloane, Nov 07 2002
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STATUS
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approved
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