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A215944 Fermat pseudoprimes to base 2 with three prime factors divisible by a smaller Fermat pseudoprime to base 2.

%I A215944 #19 Feb 16 2025 08:33:18
%S A215944 13981,137149,158369,176149,276013,285541,294409,348161,387731,423793,
%T A215944 488881,493697,617093,625921,847261,1052503,1052929,1104349,1128121,
%U A215944 1152271,1398101,1461241,1472353,1507561,1534541,1549411,1746289,1840357,1857241,2299081
%N A215944 Fermat pseudoprimes to base 2 with three prime factors divisible by a smaller Fermat pseudoprime to base 2.
%C A215944 These are members of A215150 having three divisors. - _T. D. Noe_, Aug 28 2012
%C A215944 Almost all the numbers from the sequence above can be written as p*((m + 1)*p - m)*((n + 1)*p - n), where m, n, p are natural numbers (in the brackets is written the Poulet number which every one of them is divisible by):
%C A215944 (1)  n*(2*n - 1)*(3*n - 2): the number 294409 (2701);
%C A215944 (2)  n*(2*n - 1)*(5*n - 4): the numbers 285541 (4681), 488881 (2701);
%C A215944 (3)  n*(2*n - 1)*(11*n - 10): the number 625921 (10261);
%C A215944 (4)  n*(2*n - 1)*(15*n - 14): the number 1461241 (2701);
%C A215944 (5)  n*(3*n - 2)*(4*n - 3): the numbers 13981 (341), 137149 (2047);
%C A215944 (6)  n*(3*n - 2)*(5*n - 4): the number 1152271 (5461);
%C A215944 (7)  n*(3*n - 2)*(8*n - 7): the number 1840357 (5461);
%C A215944 (8)  n*(3*n - 2)*(10*n - 9): the number 2299081 (5461);
%C A215944 (9)  n*(3*n - 2)*(12*n - 11): the number 1746289 (4033);
%C A215944 (10) n*(3*n - 2)*(31*n - 30): the number 1052503 (15709);
%C A215944 (11) n*(3*n - 2)*(102*n - 101): the number 348161 (341);
%C A215944 (12) n*(3*n - 2)*(442*n - 441): the number 1507561 (341);
%C A215944 (13) n*(4*n - 3)*(7*n - 6): the number 176149 (1387);
%C A215944 (14) n*(4*n - 3)*(11*n - 10): the number 276013 (1387);
%C A215944 (15) n*(4*n - 3)*(12*n - 11): the number 1104349 (3277);
%C A215944 (16) n*(4*n - 3)*(31*n - 30): the number 1398101 (15709);
%C A215944 (17) n*(5*n - 4)*(6*n - 5): the number 847261 (4681);
%C A215944 (18) n*(5*n - 4)*(8*n - 7): the number 1128121 (4681);
%C A215944 (19) n*(5*n - 4)*(11*n - 10): the number 1549411 (4681);
%C A215944 (20) n*(6*n - 5)*(11*n - 10): the number 1857241 (10261);
%C A215944 (21) n*(6*n - 5)*(16*n - 15): the number 423793 (4369);
%C A215944 (22) n*(7*n - 6)*(16*n - 15): the number 493697 (4369);
%C A215944 (23) n*(15*n - 14)*(16*n - 15): the number 1052929 (4369);
%C A215944 (24) n*(16*n - 15)*(21*n - 20): the number 1472353 (4369).
%C A215944 The only few numbers from the sequence above that can’t be written this way are multiples of the Poulet number 5461 and can be, instead, written as 5461*(42*k - 13): 158369 = 5461*29, 387731 = 5461*71, 617093 = 5461*113 and 1534541 = 5461*281.
%C A215944 Conjecture: The only Fermat pseudoprimes to base 2 divisible by a smaller Fermat pseudoprime to base 2 that can’t be written as p*((m + 1)*p - m)*((n + 1)*p - n), where m, n, p are natural numbers, are multiples of 5461 and can be written as 5461*(42*k - 13).
%C A215944 Conjecture is checked for the numbers from the sequence above and for the first 15 Poulet numbers with four prime factors.
%C A215944 Note: There are Fermat pseudoprimes to base 2 divisible by 5461 that can be written as p*((m + 1)*p - m)*((n + 1)*p - n); these ones can be written as 5461*(42*k + 43). Numbers from this category are: 1152271 = 5461*211, 1840357 = 5461*337, 2299081 = 5461*421.
%H A215944 Charles R Greathouse IV, <a href="/A215944/b215944.txt">Table of n, a(n) for n = 1..10000</a>
%H A215944 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>
%o A215944 (PARI) list(lim)=my(v=List(),t,psp); forprime(p=3,sqrtint(lim\3), forprime(q=p,lim\p\3, t=p*q; if(Mod(2,t)^t!=2, next); forprime(r=3, lim\t, psp=t*r; if(Mod(2,psp)^psp==2, listput(v,psp))))); Set(v) \\ _Charles R Greathouse IV_, Jun 30 2017
%Y A215944 Cf. A001567, A215150, A215672.
%K A215944 nonn
%O A215944 1,1
%A A215944 _Marius Coman_, Aug 28 2012