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A215672 Fermat pseudoprimes to base 2 with three prime factors.

%I A215672 #29 Feb 19 2025 04:05:42
%S A215672 561,645,1105,1729,1905,2465,2821,4371,6601,8481,8911,10585,12801,
%T A215672 13741,13981,15841,16705,25761,29341,30121,30889,33153,34945,41665,
%U A215672 46657,52633,57421,68101,74665,83665,87249,88561,91001,93961,113201,115921,121465,137149
%N A215672 Fermat pseudoprimes to base 2 with three prime factors.
%C A215672 Fermat pseudoprimes to base 2 are also called Poulet numbers.
%C A215672 Most of the terms shown can be written in one of the following two ways:
%C A215672   (1) p*(p*(n + 1) - n)*(p*(m + 1) - m);
%C A215672   (2) p*(p*n - (n + 1))*(p*m - (m + 1)),
%C A215672 where p is the smallest of the three prime factors and n, m natural numbers.
%C A215672 Exempli gratia for Poulet numbers from the first category:
%C A215672   10585 = 5*29*73 = 5*(5*7 - 6)*(5*18 - 17);
%C A215672   13741 = 7*13*151 = 7*(7*2 - 1)*(7*25 - 24);
%C A215672   13981 = 11*31*41 = 11*(11*3 - 2)*(11*4 - 3);
%C A215672   29341 = 13*37*61 = 13*(13*3 - 2)*(13*5 - 4);
%C A215672   137149 = 23*67*89 = 23*(23*3 - 2)*(23*4 - 3).
%C A215672 Exempli gratia for Poulet numbers from the second category:
%C A215672   6601 = 7*23*41 = 7*(7*4 - 5)*(7*7 - 8).
%C A215672 Note: from the numbers from the sequence above, just the numbers 30889, 88561 and 91001 can't be written in one of the two ways.
%C A215672 What these three numbers have in common: they all have a prime divisor q of the form 30*k + 23 (i.e. 23, 53, 83) and can be written as q*((r + 1)*q - r), where r is a natural number.
%C A215672 Conjecture: Any Poulet number P with three or more prime divisors has at least one prime divisor q for that can be written as P = q*((r + 1)*q - r), where r is a natural number.
%C A215672 Note: it can be proved that a Carmichael number can be written this way for any of its prime divisors - see the sequence A213812.
%C A215672 Note: there are also many Poulet numbers with two prime divisors that can be written this way, but here are few exceptions: 7957, 23377, 42799, 49981, 60787.
%C A215672 The conjecture fails for a(80) = 617093 = 43 * 113 * 127. - _Charles R Greathouse IV_, Dec 07 2014
%C A215672 First differs from A074380 at n=56. - _Amiram Eldar_, Jun 28 2019
%H A215672 Amiram Eldar, <a href="/A215672/b215672.txt">Table of n, a(n) for n = 1..10000</a>
%H A215672 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>
%H A215672 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>
%t A215672 Select[Range[10^5], PrimeNu[#] == 3 && PowerMod[2, (# - 1), #] == 1 &] (* _Amiram Eldar_, Jun 28 2019 *)
%o A215672 (PARI) is(n)=Mod(2,n)^n==2 && bigomega(n)==3 \\ _Charles R Greathouse IV_, Dec 07 2014
%Y A215672 The even terms form A296117.
%Y A215672 Cf. A001567, A007304, A074380, A080747, A213812.
%K A215672 nonn
%O A215672 1,1
%A A215672 _Marius Coman_, Aug 20 2012