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%I #76 Oct 30 2023 07:33:55
%S 341,1387,2047,2701,3277,4033,4369,4681,5461,7957,8321,10261,13747,
%T 14491,15709,18721,19951,23377,31417,31609,31621,35333,42799,49141,
%U 49981,60701,60787,65077,65281,80581,83333,85489,88357,90751,104653,123251,129889,130561
%N Fermat pseudoprimes to base 2 with two prime factors.
%C This sequence is the same as A050217 for the first 60 terms and starts to differ at the 61st.
%C Conjecture: For any biggest prime factor of a Poulet number p1 with two prime factors, there exists a series with infinitely many Poulet numbers p2 formed this way: p2 mod (p1 - d) = d, where d is the biggest prime factor of p1. Note: it can be seen that the Poulet numbers divisible by 73 bigger than 2701 (7957, 10585, 15841, 31609, etc.) can be written as 1314*n + 73 as well as 2628*m + 73.
%C Conjecture: Any Poulet number p2 divisible by d can be written as (p1 - d)*n + d, where n is a positive integer, if there exists a smaller Poulet number p1 with two prime factors divisible by d.
%C Note: This conjecture can't be extrapolated for Poulet numbers p1 with more than two prime factors; for instance, if p1 = 561 = 3*11*17, there indeed are bigger Poulet numbers divisible by 17 (such as 1105 and 4369) that can be written as 544*n + 17, but there also exist such numbers that can't be written this way, e.g., 2465. But the first conjecture can be extrapolated.
%C Conjecture: For any biggest prime factor of a Poulet number p1 exists a series with infinitely many Poulet numbers p2 formed this way: p2 mod (p1 - d) = d, where d is the biggest prime factor of p1.
%C For each prime p, there are only a finite number of q such that p*q is here. See A085014. Sequence A180471 lists the factors of terms of this sequence. - _T. D. Noe_, Sep 20 2012
%C Numbers n = p*q such that n divides 2^(p-1)-1 and 2^(q-1)-1, where p,q are primes; thus 2^gcd(p-1,q-1) == 1 (mod n). - _Thomas Ordowski_, Aug 27 2016
%C These are semiprimes p*q such that 2^(p+q-2) == 1 (mod p*q). Proof: 2^(p-1) == 1 (mod p) and 2^(q-1) == 1 (mod q), so 2^((p-1)*(q-1)) == 1 (mod p*q), and (p-1)*(q-1) = (p*q-1)-(p+q-2). - _Amiram Eldar_ and _Thomas Ordowski_, Apr 02 2021
%H Charles R Greathouse IV, <a href="/A214305/b214305.txt">Table of n, a(n) for n = 1..10000</a> (first 3299 terms from T. D. Noe)
%H Marius Coman, <a href="https://www.researchgate.net/profile/Marius_Coman/publication/280626311_Conjecture_that_states_that_a_Fermat_number_is_either_prime_either_divisible_by_a_2-Poulet_number/links/55bfd1f808ae9289a09b62cb.pdf">Conjecture that states that a Mersenne number with odd exponent is either prime either divisible by a 2-Poulet number</a>, 2015.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>.
%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>
%e Few examples for the first 4 Poulet numbers with two prime factors:
%e For p1 = 341 = 11*31, the following Poulet numbers p2 for which p2 mod 310 = 31 were obtained: 2821, 4371, 4681, 10261 etc.
%e For p1 = 1387 = 19*73, the following Poulet numbers p2 for which p2 mod 1314 = 73 were obtained: 2701, 7957, 10585, 15841 etc.
%e For p1 = 2047 = 23*89, the following Poulet numbers p2 for which p2 mod 1958 = 89 were obtained: 31417, 35333, 60787, 62745 etc.
%e For p1 = 2701 = 37*73, the following Poulet numbers p2 for which p2 mod 2628 = 73 were obtained: 7957, 10585, 15841 etc.
%t SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; Select[Range[200000], SemiPrimeQ[#] && PowerMod[2, #-1, #] == 1 &] (* _T. D. Noe_, Jul 12 2012 *)
%o (PARI) list(lim)=my(v=List());forprime(p=31,lim\11, forprime(q=11,min(p-1,lim\p), if(Mod(2,p)^(q-1)==1 && Mod(2,q)^(p-1)==1, listput(v,p*q)))); if(lim>=1093^2,listput(v,1093^2)); if(lim>=3511^2,listput(v,3511^2)); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Sep 20 2012
%Y Subsequence of A050217.
%Y Cf. A001567.
%K nonn
%O 1,1
%A _Marius Coman_, Jul 12 2012
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