A351336
Odd pseudoprimes to base 3; composite terms of A271116.
Original entry on oeis.org
91, 121, 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, 24661, 24727, 28009, 29161, 29341, 30857, 31621
Offset: 1
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q[n_] := CompositeQ[n] && Divisible[Round[3^n/12], n]; Select[Range[32000], q] (* Amiram Eldar, Feb 09 2022 *)
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is(n) = (n>1) && !isprime(n) && (lift(Mod(3, 4*n)^(n-1))==1); \\ Michel Marcus, Feb 09 2022; after A271116
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list(lim)=my(v=List()); forcomposite(n=91,lim\1, if(bittest(34,n%6) && Mod(3,n)^(n-1)==1, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Feb 09 2022
A328662
Super pseudoprimes (or superpseudoprimes) to base 3: Fermat pseudoprimes to base 3 all of whose divisors that are larger than 1 are either primes or Fermat pseudoprimes to base 3.
Original entry on oeis.org
91, 121, 671, 703, 949, 1541, 1891, 2701, 3281, 7381, 8401, 12403, 14383, 15203, 16531, 18721, 23521, 24727, 28009, 30857, 31621, 31697, 38503, 44287, 46999, 47197, 49051, 49141, 55261, 55969, 63139, 72041, 74593, 79003, 82513, 83333, 88573, 88831, 90751, 96139
Offset: 1
91 is in the sequence since it is a Fermat pseudoprime to base 3, and its proper divisors that are larger than 1 are the primes 7 and 13.
7381 is in the sequence since it is a Fermat pseudoprime to base 3, and its proper divisors that are larger than 1 are the primes 11 and 61, and the composite numbers 121 and 671 that are Fermat pseudoprimes to base 3.
- Michal Krížek, Florian Luca, and Lawrence Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Springer-Verlag, New York, 2001, chapter 12, Fermat's Little Theorem, Pseudoprimes, and Superpseudoprimes, pp. 130-146.
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- J. Fehér and P. Kiss, Note on super pseudoprime numbers, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., Vol. 26 (1983), pp. 157-159, entire volume.
- B. M. Phong, On super pseudoprimes which are products of three primes, Ann. Univ. Sci. Budapest. Eótvós Sect. Math., Vol. 30 (1987), pp. 125-129, entire volume.
- Andrzej Rotkiewicz, Solved and unsolved problems on pseudoprime numbers and their generalizations, Applications of Fibonacci numbers, Springer, Dordrecht, 1999, pp. 293-306.
- Lawrence Somer, On superpseudoprimes, Mathematica Slovaca, Vol. 54, No. 5 (2004), pp. 443-451.
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aQ[n_]:= CompositeQ[n] && AllTrue[Rest[Divisors[n]], PowerMod[3, #-1, #] == 1 &]; Select[Range[10^5], aQ]
Showing 1-2 of 2 results.
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