Search: id:A001567
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%I A001567 M5441 N2365
%S A001567 341,561,645,1105,1387,1729,1905,2047,2465,2701,2821,3277,4033,4369,4371,
%T A001567 4681,5461,6601,7957,8321,8481,8911,10261,10585,11305,12801,13741,13747,
%U A001567 13981,14491,15709,15841,16705,18705,18721,19951,23001,23377,25761,29341
%N A001567 Pseudoprimes, also called Sarrus numbers: pseudoprimes to base 2.
%C A001567 An odd composite number n is a Fermat pseudoprime to base b iff b^(n-1) 
               == 1 mod n. Fermat pseudoprimes to base 2 are often simply called 
               pseudoprimes.
%C A001567 Theorem: If both numbers q and 2q-1 are primes (q is in the sequence 
               A005382) and n=q*(2q-1) then 2^(n-1)==1 (mod n) (n is in the sequence) 
               iff q is of the form 12k+1. 2701,18721,49141,104653,226801,665281,
               721801,... is the related subsequence. This subsequence is also a 
               subsequence of the sequences A005937 and A020137. - Farideh Firoozbakht 
               (mymontain(AT)yahoo.com), Sep 15 2006
%C A001567 Also composite numbers n such that n divides 2^n - 2. It is known that 
               all primes p divide 2^(p-1) - 1. There are only two known numbers 
               n such that n^2 divides 2^(n-1) - 1, A001220(n) = {1093, 3511} Wieferich 
               primes p: p^2 divides 2^(p-1) - 1. 1093^2 and 3511^2 are the terms 
               of a(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 06 2006
%C A001567 An odd composite number 2n+1 is in the sequence iff multiplicative order 
               of 2 (mod 2n+1) divides 2n. - Ray Chandler (rayjchandler(AT)sbcglobal.net), 
               May 26 2008
%C A001567 Contribution from Artur Jasinski (grafix(AT)csl.pl), Dec 28 2008: (Start)
%C A001567 The Carmichael numbers A002997 are subset of this sequence.
%C A001567 For the Sarrus numbers which are not Carmichael numbers see A153508. 
               (End)
%D A001567 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001567 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001567 R. K. Guy, Unsolved Problems Theory of Numbers, A12.
%D A001567 D. H. Lehmer, Errata for Poulet's table, Math. Comp., 25 (1971), 944-945. 
               25 944 1971.
%D A001567 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 
               105, National Research Council, Washington, DC, 1941, p. 48.
%D A001567 P. Poulet, Tables des nombres composes verifiant le theoreme du Fermat 
               pour le module 2 jusqu'a 100.000.000, Sphinx (Brussels), 8 (1938), 
               42-45.
%D A001567 W. Sierpi\'{n}ski, Elementary Theory of Numbers. Pa\'{n}st. Wydaw. Nauk., 
               Warsaw, 1964, p. 215.
%H A001567 N. J. A. Sloane, <a href="b001567.txt">Table of n, a(n) for n = 1..101629</
               a> [The pseudoprimes up to 10^12, from Richard Pinch's web site - 
               see links below]
%H A001567 J. Bernheiden, <a href="http://www.mathe-schule.de/download/pdf/Primzahl/
               PSP.pdf">Pseudoprimes (Text in German)</a>
%H A001567 F. Di Noto and A. R. Tulumello, <a href="http://www.geocities.com/g_armillotta/
               metodo19/di_noto13.html">Nuovo test di primalita</a>.
%H A001567 G. P. Michon, <a href="http://home.att.net/~numericana/answer/pseudo.htm">
               Pseudoprimes</a>
%H A001567 Richard Pinch, <a href="http://www.chalcedon.demon.co.uk/rgep/carpsp.html">
               Pseudoprimes</a>
%H A001567 F. Richman, <a href="http://www.math.fau.edu/Richman/carm.htm">Primality 
               testing with Fermat's little theorem</a>
%H A001567 W. Sierpi\'{n}ski, <a href="http://matwbn.icm.edu.pl/kstresc.php?tom=42&wyd=10">
               Elementary Theory of Numbers</a>, Warszawa 1964.
%H A001567 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PouletNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A001567 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Pseudoprime.html">Link to a section of The World of Mathematics.</
               a>
%H A001567 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FermatPseudoprime.html">Fermat Pseudoprime</a>
%H A001567 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pseudoprime">Pseudoprime</
               a>
%H A001567 <a href="Sindx_Ps.html#pseudoprimes">Index entries for sequences related 
               to pseudoprimes</a>
%F A001567 Theorem: If both numbers q and 2q-1 are primes and n=q*(2q-1) then n 
               is a pseudoprime in base 2 (n is in the sequence A001567 ) iff q 
               == 1 (mod 4). - Farideh Firoozbakht (mymontain(AT)yahoo.com), Sep 
               11 2006
%F A001567 Equivalently: Theorem: If both numbers q and 2q-1 are primes and n=q*(2q-1) 
               then n is a pseudoprime to base 2 (n is in the sequence A001567 ) 
               iff q == 1 (mod 12). - Farideh Firoozbakht (mymontain(AT)yahoo.com), 
               Sep 15 2006
%t A001567 Select[Range[4100], ! PrimeQ[ # ] && PowerMod[2, (# - 1), # ] == 1 &] 
               - Farideh Firoozbakht (mymontain(AT)yahoo.com), Sep 15 2006
%o A001567 (PARI) q=1;vector(50,i,until( !isprime(q) & (1<<(q-1)-1)%q == 0, q+=2);
               q) [M. F. Hasler, May 04 2007]
%Y A001567 Cf. A002997, A052155, A083737, A084653, A005382, A005937, A020137.
%Y A001567 Cf. A001220 = Wieferich primes p: p^2 divides 2^(p-1) - 1.
%Y A001567 A153508 [From Artur Jasinski (grafix(AT)csl.pl), Dec 28 2008]
%Y A001567 Sequence in context: A020188 A025353 A025345 this_sequence A006970 A007324 
               A007011
%Y A001567 Adjacent sequences: A001564 A001565 A001566 this_sequence A001568 A001569 
               A001570
%K A001567 nonn,nice
%O A001567 1,1
%A A001567 N. J. A. Sloane (njas(AT)research.att.com).
%E A001567 More terms from David W. Wilson Aug 15 1996.

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