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A076336
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(Provable) Sierpinski numbers: odd n such that for all k >= 1 the numbers n*2^k + 1 are composite.
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14
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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. 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).
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REFERENCES
| 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.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..13394 (from McLean with duplicates removed)
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
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CROSSREFS
| Cf. A076337, A076335, A003261, A052333, A101036.
Sequence in context: A017587 A038826 A038815 * A123159 A184230 A186612
Adjacent sequences: A076333 A076334 A076335 * A076337 A076338 A076339
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KEYWORD
| nonn,hard,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Nov 07 2002
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