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 A005935 Pseudoprimes to base 3. (Formerly M5362) 40
 91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7381, 8401, 8911, 10585, 11011, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345, 23521, 24046, 24661, 24727, 28009, 29161 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Theorem: If q>3 and both numbers q and (2q-1) are primes then n=q*(2q-1) is a pseudoprime to base 3 (i.e. n is in the sequence). So for n>2, A005382(n)*(2*A005382(n)-1) is in the sequence (see Comments lines for the sequence A122780). 91,703,1891,2701,12403,18721,38503,49141... are such terms. This sequence is a subsequence of A122780. - Farideh Firoozbakht, Sep 13 2006 Composite numbers n such that 3^(n-1) == 1 (mod n). Theorem (R. Steuerwald, 1948): if n is a pseudoprime to base b and gcd(n,b-1)=1, then (b^n-1)/(b-1) is a pseudoprime to base b. In particular, if n is an odd pseudoprime to base 3, then (3^n-1)/2 is a pseudoprime to base 3. - Thomas Ordowski, Apr 06 2016 Steuerwald's theorem can be strengthened by weakening his assumption as follows: if n is a weak pseudoprime to base b and gcd(n,b-1)=1, then ... - Thomas Ordowski, Feb 23 2021 REFERENCES J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 91, p. 33, Ellipses, Paris 2008. R. K. Guy, Unsolved Problems in Number Theory, A12. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS R. J. Mathar, T. D. Noe and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..102839 (terms a(1)-a(798) from R. J. Mathar, a(799)-a(1000) from T. D. Noe) J. Bernheiden, Pseudoprimes (Text in German) C. Pomerance & N. J. A. Sloane, Correspondence, 1991 F. Richman, Primality testing with Fermat's little theorem Rudolf Steuerwald, Über die Kongruenz a^(n-1) == 1 (mod n), Sitzungsber. math.-naturw. Kl. Bayer. Akad. Wiss. München, 1948, pp. 69-70. Eric Weisstein's World of Mathematics, Fermat Pseudoprime MATHEMATICA base = 3; t = {}; n = 1; While[Length[t] < 100, n++; If[! PrimeQ[n] && PowerMod[base, n-1, n] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Feb 21 2012 *) PROG (PARI) is_A005935(n)={Mod(3, n)^(n-1)==1 & !ispseudoprime(n) & n>1}  \\ M. F. Hasler, Jul 19 2012 CROSSREFS Pseudoprimes to other bases: A001567 (2), A005936 (5), A005937 (6), A005938 (7), A005939 (10). Cf. A005382, A122780. Sequence in context: A236845 A157345 A092125 * A020307 A328662 A020235 Adjacent sequences:  A005932 A005933 A005934 * A005936 A005937 A005938 KEYWORD nonn AUTHOR EXTENSIONS More terms from David W. Wilson, Aug 15 1996 STATUS approved

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Last modified April 16 07:59 EDT 2021. Contains 343030 sequences. (Running on oeis4.)