A052155 -id:A052155 - cp's OEIS frontend

A001567 Fermat pseudoprimes to base 2, also called Sarrus numbers or Poulet numbers.

341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, 10261, 10585, 11305, 12801, 13741, 13747, 13981, 14491, 15709, 15841, 16705, 18705, 18721, 19951, 23001, 23377, 25761, 29341

Offset: 1

N. J. A. Sloane

A composite number n is a Fermat pseudoprime to base b if and only if b^(n-1) == 1 (mod n). Fermat pseudoprimes to base 2 are often simply called pseudoprimes.
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) if and only if q is of the form 12k + 1. The sequence 2701, 18721, 49141, 104653, 226801, 665281, 721801, ... is related. This subsequence is also a subsequence of the sequences A005937 and A020137. - Farideh Firoozbakht, Sep 15 2006
Also, composite odd numbers n such that n divides 2^n - 2 (cf. A006935). 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, Nov 06 2006
An odd composite number 2n + 1 is in the sequence if and only if multiplicative order of 2 (mod 2n+1) divides 2n. - Ray Chandler, May 26 2008
The Carmichael numbers A002997 are a subset of this sequence. For the Sarrus numbers which are not Carmichael numbers, see A153508. - Artur Jasinski, Dec 28 2008
An odd number n greater than 1 is a Fermat pseudoprime to base b if and only if ((n-1)! - 1)*b^(n-1) == -1 (mod n). - Arkadiusz Wesolowski, Aug 20 2012
The name "Sarrus numbers" is after Frédéric Sarrus, who, in 1819, discovered that 341 is a counterexample to the "Chinese hypothesis" that n is prime if and only if 2^n is congruent to 2 (mod n). - Alonso del Arte, Apr 28 2013
The name "Poulet numbers" appears in Monografie Matematyczne 42 from 1932, apparently because Poulet in 1928 produced a list of these numbers (cf. Miller, 1975). - Felix Fröhlich, Aug 18 2014
Numbers n > 2 such that (n-1)! + 2^(n-1) == 1 (mod n). Composite numbers n such that (n-2)^(n-1) == 1 (mod n). - Thomas Ordowski, Feb 20 2016
The only twin pseudoprimes up to 10^13 are 4369, 4371. - Thomas Ordowski, Feb 12 2016
Theorem (A. Rotkiewicz, 1965): (2^p-1)*(2^q-1) is a pseudoprime if and only if p*q is a pseudoprime, where p,q are different primes. - Thomas Ordowski, Mar 31 2016
Theorem (W. Sierpiński, 1947): if n is a pseudoprime (odd or even), then 2^n-1 is a pseudoprime. - Thomas Ordowski, Apr 01 2016
If 2^n-1 is a pseudoprime, then n is a prime or a pseudoprime (odd or even). - Thomas Ordowski, Sep 05 2016
From Amiram Eldar, Jun 19 2021, Apr 21 2024: (Start)
Erdős (1950) called these numbers "almost primes".
According to Erdős (1949) and Piza (1954), the term "pseudoprime" was coined by D. H. Lehmer.
Named after the French mathematician Pierre de Fermat (1607-1665), or, alternatively, after the Belgian mathematician Paul Poulet (1887-1946), or, the French mathematician Pierre Frédéric Sarrus (1798-1861). (End)
If m is a term of this sequence, then (m-1)/ord(2,m) >= 5, where ord(a,m) is the multiplicative order of a modulo m; see my link below. Actually, it seems that we always have (m-1)/ord(2,m) >= 9. - Jianing Song, Nov 04 2024

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 80.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A12, pp. 44-50.
George P. Loweke, The Lore of Prime Numbers. New York: Vantage Press (1982), p. 22.
Øystein Ore, Number Theory and Its History, McGraw-Hill, 1948.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 88-92.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 145.

N. J. A. Sloane, Table of n, a(n) for n = 1..101629 [The pseudoprimes up to 10^12, from Richard Pinch's web site - see links below]
Jonathan Bayless and Paul Kinlaw, Explicit Bounds for the Sum of Reciprocals of Pseudoprimes and Carmichael Numbers, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.4.
Jens Bernheiden, Pseudoprimes (Text in German).
John Brillhart, N. J. A. Sloane, J. D. Swift, Correspondence, 1972.
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
Paul Erdős, On the converse of Fermat's theorem, The American Mathematical Monthly, Vol. 56, No. 9 (1949), pp. 623-624; alternative link.
Paul Erdős, On almost primes, Amer. Math. Monthly, Vol. 57, No. 6 (1950), pp. 404-407; alternative link.
Jan Feitsma, The pseudoprimes below 2^64.
William Galway, Tables of pseudoprimes and related data [Includes a file with pseudoprimes up to 2^64.]
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly, Vol. 95, No. 8 (1988), pp. 697-712. [Annotated scanned copy]
Paul Kinlaw, The reciprocal sums of pseudoprimes and Carmichael numbers, Mathematics of Computation (2023).
D. H. Lehmer, Guide to Tables in the Theory of Numbers, Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 48.
D. H. Lehmer, Errata for Poulet's table, Math. Comp., Vol. 25, No. 116 (1971), pp. 944-945.
D. H. Lehmer, Errata for Poulet's table. [annotated scanned copy]
Gérard P. Michon, Pseudoprimes.
J. C. P. Miller, On factorization, with a suggested new approach, Math. Comp., Vol. 29, No. 129 (1975), pp. 155-172. - _Felix Fröhlich_, Aug 18 2014
Robert Morris, Some observations on the converse of Fermat's theorem, unpublished memorandum, Oct 03 1973.
Richard Pinch, Pseudoprimes.
P. A. Piza, Fermat Coefficients, Mathematics Magazine, Vol. 27, No. 3 (1954), pp. 141-146.
Carl Pomerance & N. J. A. Sloane, Correspondence, 1991.
Paul Poulet, Tables des nombres composés vérifiant le théorème de Fermat pour le module 2 jusqu'à 100.000.000, Sphinx (Brussels), Vol. 8 (1938), pp. 42-45. [annotated scanned copy]
Fred Richman, Primality testing with Fermat's little theorem.
Andrzej Rotkiewicz, Sur les nombres pseudopremiers de la forme MpMq, Elemente der Mathematik, Vol. 20 (1965), pp. 108-109.
Waclaw Sierpiński, Remarque sur une hypothèse des Chinois concernant les nombres (2^n-2)/n, Colloquium Mathematicum, Vol. 1 (1947), p. 9.
Waclaw Sierpiński, Elementary Theory of Numbers, Państ. Wydaw. Nauk., Warszawa, 1964, p. 215.
Jianing Song, Notes on Fermat Pseudoprimes.
Eric Weisstein's World of Mathematics, Chinese Hypothesis, Fermat Pseudoprime, Poulet Number, and Pseudoprime.
Wikipedia, Chinese hypothesis and Pseudoprime.
Index entries for sequences related to pseudoprimes

Cf. A002997, A005382, A020137, A052155, A083737, A084653, A153508.
Cf. A001220 = Wieferich primes p: p^2 divides 2^(p-1) - 1.
Cf. A005935, A005936, A005937, A005938, A005939, A020136-A020228 (pseudoprimes to bases 3 through 100).

[n: n in [3..3*10^4 by 2] | IsOne(Modexp(2,n-1,n)) and not IsPrime(n)]; // Bruno Berselli, Jan 17 2013

select(t -> not isprime(t) and 2 &^(t-1) mod t = 1, [seq(i,i=3..10^5,2)]); # Robert Israel, Feb 18 2016

Select[Range[3,30000,2], ! PrimeQ[ # ] && PowerMod[2, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)

q=1;vector(50,i,until( !isprime(q) & (1<<(q-1)-1)%q == 0, q+=2);q) \\ M. F. Hasler, May 04 2007

is_A001567(n)={Mod(2,n)^(n-1)==1 && !isprime(n) && n>1}  \\ M. F. Hasler, Oct 07 2012, updated to current PARI syntax and to exclude even pseudoprimes on Mar 01 2019

Sum_{n>=1} 1/a(n) is in the interval (0.015260, 33) (Bayless and Kinlaw, 2017). The upper bound was reduced to 0.0911 by Kinlaw (2023). - Amiram Eldar, Oct 15 2020, Feb 24 2024

More terms from David W. Wilson, Aug 15 1996
Replacement of broken geocities link by Jason G. Wurtzel, Sep 05 2010
"Poulet numbers" added to name by Joerg Arndt, Aug 18 2014

A083737 Pseudoprimes to bases 2, 3 and 5.

Original entry on oeis.org

1729, 2821, 6601, 8911, 15841, 29341, 41041, 46657, 52633, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 670033, 721801, 748657

Offset: 1

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

a(n) = n-th positive integer k(>1) such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k)
See A153580 for numbers k > 1 such that 2^k-2, 3^k-3 and 5^k-5 are all divisible by k but k is not a Carmichael number (A002997).
Note that a(1)=1729 is the Hardy-Ramanujan number. - Omar E. Pol, Jan 18 2009

			a(1)=1729 since it is the first number such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 102 from R. J. Mathar)
J. Bernheiden, Pseudoprimes (Text in German)
F. Richman, Primality testing with Fermat's little theorem
Index entries for sequences related to pseudoprimes

Proper subset of A052155. Superset of A230722. Cf. A153580, A002997, A001235, A011541.

Select[ Range[838200], !PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, 1 - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 & ]

is(n)=!isprime(n)&&Mod(2,n)^(n-1)==1&&Mod(3,n)^(n-1)==1&&Mod(5,n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012

Edited by Robert G. Wilson v, May 06 2003

Edited by N. J. A. Sloane, Jan 14 2009

A083876 Least pseudoprime to base 2 through base prime(n).

Original entry on oeis.org

341, 1105, 1729, 29341, 29341, 162401, 252601, 252601, 252601, 252601, 252601, 252601, 1152271, 2508013, 2508013, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 6733693, 6733693, 6733693

Offset: 1

Robert G. Wilson v, May 06 2003

nonn

Records: 341, 1105, 1729, 29341, 162401, 252601, 1152271, 2508013, 3828001, 6733693, 17098369, 17236801, 29111881, 82929001, 172947529, 216821881, 228842209, 366652201, .... - Robert G. Wilson v, May 11 2012
Conjecture: for n > 1, a(n) is the smallest Carmichael number k with lpf(k) > prime(n). It seems that such Carmichael numbers have exactly three prime factors. - Thomas Ordowski, Apr 18 2017
The conjecture is true if a(n) < A285549(n) for all n > 1. It holds for all a(n) < 2^64. - Max Alekseyev and Thomas Ordowski, Mar 13 2018
If prime(n) < m < a(n), then m is prime if and only if p^(m-1) == 1 (mod m) for every prime p <= prime(n). - Thomas Ordowski, Mar 05 2018
By this conjecture in the second comment, a(n) <= A135720(n+1), with equality for n > 1 iff a(n) < a(n+1), namely for n = 2, 3, 5, 6, 12, 13, 15, 25, 28, 29, ... For such n, a(n) gives all terms of A300629 > 561. - Thomas Ordowski, Mar 10 2018

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 100 terms from Robert G. Wilson v)

Cf. A002997, A001567, A052155, A083737, A087788, A083739, A135720, A141768, A250199, A271221, A285549, A300629.

k = 4; Do[l = Table[ Prime[i], {i, 1, n}]; While[ PrimeQ[k] || Union[PowerMod[l, k - 1, k]] != {1}, k++ ]; Print[k], {n, 1, 29}]

isps(k, n) = {if (isprime(k), return (0)); my(nbok = 0); for (b=2, prime(n), if (Mod(b, k)^(k-1) == 1, nbok++, break)); if (nbok==prime(n)-1, return (1));}
a(n) = {my(k=2); while (!isps(k, n), k++); return (k);} \\ Michel Marcus, Apr 27 2018

A271221 Smallest Fermat pseudoprime k to all bases b = 2, 3, 4, ..., n.

Original entry on oeis.org

341, 1105, 1105, 1729, 1729, 29341, 29341, 29341, 29341, 29341, 29341, 162401, 162401, 162401, 162401, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601

Offset: 2

Felix Fröhlich, Apr 02 2016

nonn

a(n) is the smallest composite k such that b^(k-1) == 1 (mod (b-1)k) for every b = 2, 3, 4, ..., n. For more comments, see A083876 and A300629. - Max Alekseyev and Thomas Ordowski, Apr 29 2018

Cf. A052155, A083737, A083739, A083876, A300629.

a(n) = forcomposite(c=1, , my(i=0); for(b=2, n, if(Mod(b, c)^(c-1)==1, i++)); if(i==n-1, return(c)));

Edited by Thomas Ordowski, Apr 29 2018

Corrected a typo within the initial terms by Jens Ahlström, Apr 23 2024

A114246 Number of Fermat pseudoprimes to bases 2 and 3 less than 10^n.

Original entry on oeis.org

0, 0, 0, 7, 23, 66, 187, 485, 1272, 3270, 8139, 20015, 50090, 125284, 318729, 813380, 2104139, 5514763, 14627441

Offset: 1

Eric W. Weisstein, Nov 18 2005

Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Index entries for sequences related to pseudoprimes.

Cf. A052155.

Table[Count[Select[Range[2, 10^6], ! PrimeQ[#] && PowerMod[2, # - 1, #] == 1 && PowerMod[3, # - 1, #] == 1 &], x_ /; x < 10^n], {n, 6}]  (* Robert Price, Jun 09 2019 *)

my(n=1, i=0); forcomposite(k=1, 1e20, if(Mod(2, k)^(k-1)==1 && Mod(3, k)^(k-1)==1, i++); if(k==10^n, print1(i, ", "); n++)) \\ Felix Fröhlich, Oct 11 2015

a(9) from Felix Fröhlich, Oct 11 2015
a(10)-a(11) from Amiram Eldar, Sep 18 2021
a(12)-a(19) from Amiram Eldar, Apr 22 2022

A318761 Composite k that divides 2^(k-2) + 3^(k-2) + 6^(k-2) - 1.

Original entry on oeis.org

4, 6, 8, 12, 24, 25, 125, 174, 228, 276, 325, 348, 451, 1032, 1105, 1128, 1729, 2388, 2465, 2701, 2821, 3721, 5272, 5365, 6601, 8911, 10585, 12025, 12673, 15841, 18721, 22681, 23585, 23725, 29341, 31621, 32376, 35016, 35425, 41041, 41125, 46632, 46657, 47125

Offset: 1

Jianing Song, Sep 02 2018

nonn

Note that for primes p >= 5, p always divides 2^(p-2) + 3^(p-2) + 6^(p-2) - 1 (see A318760).

It's interesting to study the squares of primes in this sequence. For primes p >= 5, x^(p^2-2) == x^(p-2) (mod p^2) for any integer x, so p^2 is a term if and only if p^2 divides 2^(p-2) + 3^(p-2) + 6^(p-2) - 1. It's easy to see that for any prime p, p^2 is a term of this sequence if and only if p is in A238201 and p != 3 (p = 2, 5, 61, 1680023, 7308036881, there are no others up to 7*10^10). - Jianing Song, Dec 25 2018

			(2^10 + 3^10 + 6^10 - 1)/12 = 5403854 which is an integer, so 12 is a term.
(2^22 + 3^22 + 6^22 - 1)/24 = 5484238967813377 which is also an integer, so 24 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..4127 (terms below 10^10)

Cf. A238201, A318760.

A052155 is a proper subsequence.

Select[Range[48000], CompositeQ[#] && Mod[Sum[PowerMod[k, #-2, #],{k, {2, 3, 6}}], #] == 1 &] (* Amiram Eldar, Jul 17 2024 *)

b(n) = lift(Mod(2, n)^(n-2) + Mod(3, n)^(n-2) + Mod(6, n)^(n-2));
for(n=2, 30000, if(isprime(n)==0&&b(n)==1, print1(n, ", ")))

A330208 Chebyshev pseudoprimes to both base 2 and base 3: composite numbers k such that T(k, 2) == 2 (mod k) and T(k, 3) == 3 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.

Original entry on oeis.org

5719, 6061, 11395, 15841, 17119, 18721, 31535, 67199, 73555, 84419, 117215, 133399, 133951, 174021, 181259, 194833, 226801, 273239, 362881, 469201, 516559, 522899, 534061, 588455, 665281, 700321, 721801, 778261, 903959, 1162349, 1561439, 1708901, 1755001, 1809697

Offset: 1

Amiram Eldar, Dec 05 2019

nonn

Bang proved that T(p, a) == a (mod p) for every a > 0 and every odd prime. Rayes et al. (1999) defined Chebyshev pseudoprimes to base a as composite numbers k such that T(k, a) == a (mod k). They noted that there are no Chebyshev pseudoprimes in both bases 2 and 3 below 2000.

			5719 is in the sequence since 5719 = 7 * 19 * 43 is composite and both T(5719 , 2) - 2 and T(5719, 3) - 3 are divisible by 5719.

Amiram Eldar, Table of n, a(n) for n = 1..73 (terms below 10^7)
Thøger Bang, Congruence properties of Tchebycheff polynomials, Mathematica Scandinavica, Vol. 2, No. 2 (1955), pp. 327-333, alternative link,
Mohamed O. Rayes, Vilmar Trevisan, and Paul S. Wangy, Chebyshev Polynomials and Primality Tests, ICM Technical Report, Kent State University, Kent, Ohio, 1999. See page 8.

Intersection of A330206 and A330207.

Cf. A052155, A053120, A175530.

Select[Range[2*10^4], CompositeQ[#] && Divisible[ChebyshevT[#, 2] - 2, #] && Divisible[ChebyshevT[#, 3] - 3, #] &]

A333130 Numbers that are super pseudoprimes to both bases 2 and 3.

Original entry on oeis.org

2701, 18721, 31621, 49141, 83333, 90751, 104653, 226801, 282133, 653333, 665281, 721801, 873181, 1373653, 1530787, 1537381, 1584133, 1690501, 1755001, 1987021, 2008597, 2035153, 2284453, 2746589, 2944261, 3059101, 3116107, 3363121, 3375041, 3375487, 4082653, 4314967

Offset: 1

Amiram Eldar, Mar 08 2020

nonn

The first term that has more than 2 prime factors is a(1067) = A333131(1) = 11500521553.

The first term that is also a Carmichael number is a(1131) = 13079177569.

			2701 is a term since it is a Fermat pseudoprime to both bases 2 and 3, and its proper divisors that are larger than 1 are all primes: 37 and 73.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A050217 and A328662.

Subsequence of A001567, A005935, A052155 and A153513.

pspQ[n_] := CompositeQ[n] && AllTrue[Rest @ Divisors[n], PowerMod[2, # - 1, #] == 1 && PowerMod[3, # - 1, #] == 1 &]; Select[Range[10^6], pspQ]

A333131 Super pseudoprimes to both bases 2 and 3 (A333130) with more than two prime factors (counted with multiplicity).

Original entry on oeis.org

11500521553, 13079177569, 52474339009, 168003672409, 229352039821, 280792563977, 318289021201, 428178002569, 918660756421, 2015841188197, 2367478228501, 2544457029601, 2639665216117, 3023595814801, 3457449931321, 3712164285421, 4348114583017, 6046196043229

Offset: 1

Amiram Eldar, Mar 08 2020

nonn

Up to 2^64 all the 1085 terms are nonsquarefree, 2 terms have 4 prime factors: a(163) = 18362297383286473 = 3037 * 6073 * 9109 * 109297 and a(651) = 2587580959818925201 = 18121 * 36241 * 54361 * 72481, and no term have more than 4 prime factors.

			11500521553 is a term since it is a Fermat pseudoprime to both bases 2 and 3, and its proper divisors that are larger than 1 are either primes (937, 1873, 6553) or Fermat pseudoprimes to both bases 2 and 3 (1755001, 6140161, 12273769, 11500521553).

Amiram Eldar, Table of n, a(n) for n = 1..1085 (terms below 2^64)

Subsequence of A001567, A005935, A050217, A052155, A153513, A178997, A328662, A328663, A333130.

pspQ[n_] := PrimeOmega[n] > 2 && AllTrue[Rest @ Divisors[n], PowerMod[2, # - 1, #] == 1 && PowerMod[3, # - 1, #] == 1 &]; seq = {}; Do[If[pspQ[n], AppendTo[seq, n]], {n, 1, 6*10^10}]; seq

A285300 Numbers k such that 3^(k-1) == 2^(k-1) !== 1 (mod k).

Original entry on oeis.org

65, 133, 529, 793, 1649, 2059, 2321, 4187, 5185, 6305, 6541, 6697, 6817, 7471, 7613, 8113, 10963, 11521, 13213, 13333, 13427, 14701, 14981, 19171, 19201, 19909, 21349, 21667, 22177, 26065, 26467, 32873, 35443, 36569, 37333, 38897, 42121, 42127, 44023, 47081

Offset: 1

Thomas Ordowski, Apr 16 2017

nonn

All terms are odd composite numbers. There are no pseudoprimes to bases 2 or 3 in this sequence.
Are there infinitely many numbers of this kind?
From Max Alekseyev, Apr 16 2017: (Start)
Also, Fermat pseudoprimes base 2/3 that are not Fermat pseudoprimes base 2.
Also, the set difference of A073631 and either of ({1} U A001567), ({1} U A005935), or ({1} U A052155). (End)

			2^64 = 18446744073709551616 = 65 * 283796062672454640 + 16 and 3^64 = 3433683820292512484657849089281 = 65 * 52825904927577115148582293681 + 16. Therefore 65 is in the sequence.
Note: a(3) = 529 = 23^2 and a(40) = 47081 = 23^2 * 89.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A001567, A005935, A052155, A073631.

filter:= proc(n) local t;
  t:= 3 &^(n-1) mod n;
  if t = 1 then return false fi;
  t = 2 &^(n-1) mod n;
end proc:
select(filter, [seq(i,i=3..10^5,2)]); # Robert Israel, Apr 27 2017

Select[Range[2, 10^5], PowerMod[2, # - 1, #] == PowerMod[3, # - 1, #] != 1 &] (* Giovanni Resta, Apr 16 2017 *)

is(n) = Mod(3, n)^(n-1)==2^(n-1) && Mod(2, n)^(n-1)!=1 \\ Felix Fröhlich, Apr 27 2017

More terms from Giovanni Resta, Apr 16 2017

cp's OEIS Frontend

A001567 Fermat pseudoprimes to base 2, also called Sarrus numbers or Poulet numbers.

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A083737 Pseudoprimes to bases 2, 3 and 5.

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A083876 Least pseudoprime to base 2 through base prime(n).

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A271221 Smallest Fermat pseudoprime k to all bases b = 2, 3, 4, ..., n.

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A114246 Number of Fermat pseudoprimes to bases 2 and 3 less than 10^n.

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A318761 Composite k that divides 2^(k-2) + 3^(k-2) + 6^(k-2) - 1.

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A330208 Chebyshev pseudoprimes to both base 2 and base 3: composite numbers k such that T(k, 2) == 2 (mod k) and T(k, 3) == 3 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.

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A333130 Numbers that are super pseudoprimes to both bases 2 and 3.