A158358 - OEIS

%I #52 Feb 11 2019 12:17:27

%S 1194649,12327121,3914864773,5654273717,6523978189,22178658685,

%T 26092328809,31310555641,41747009305,53053167441,58706246509,

%U 74795779241,85667085141,129816911251,237865367741,259621495381,333967711897,346157884801,467032496113,575310702877,601401837037,605767053061

%N Pseudoprimes to base 2 that are not squarefree, including the even pseudoprimes.

%C Intersection of (A001567 U A006935) and A013929. Also, intersection of A015919 and A013929.

%C The first six terms are given by Ribenboim, who references calculations by Lehmer and by Pomerance, Selfridge & Wagstaff supporting "that the only possible factors p^2 (where p is a prime less than 6*10^9) of any pseudoprime, must be 1093 or 3511." Ribenboim states that the first four terms are strong pseudoprimes. The first two terms are squares of these Wieferich primes, 1093^2 and 3511^2.

%C Only Wieferich primes (A001220) can appear with an exponent greater than one. In particular, all members of this sequence are divisible by a square of a Wieferich prime. Up to 67 * 10^14 the only Wieferich primes are 1093 and 3511. - _Charles R Greathouse IV_, Sep 12 2012

%C The first term divisible by the squares of two (Wieferich) primes is a(11870) = 4578627124156945861 = 29 * 71 * 151 * 1093^2 * 3511^2. See A219346. - _Charles R Greathouse IV_, Sep 20 2012

%C Unless there are other Wieferich primes besides 1093 and 3511, the sequence is the union of A247830 and A247831. - _Max Alekseyev_, Nov 26 2017

%C The even terms are listed in A295740. - _Max Alekseyev_, Nov 26 2017 [Their indices in this sequence are 2882, 3476, 3573, 4692, 5434, 5581, 6332, 8349, 8681, 9515, ... - _Jianing Song_, Feb 08 2019]

%D P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, NY, 1991, pp. 77, 83, 167.

%H Charles R Greathouse IV, <a href="/A158358/b158358.txt">Table of n, a(n) for n = 1..10010</a> (The even terms inserted by Jianing Song)

%H Jan Feitsma, <a href="http://www.janfeitsma.nl/math/psp2/wieferich">W search: non-squarefree pseudoprimes</a>

%H R. G. E. Pinch, <a href="http://dx.doi.org/10.1007/10722028_30">The pseudoprimes up to 10^13</a>, Lecture Notes in Computer Science, 1838 (2000), 459-473. - _Felix Fröhlich_, Apr 16 2014

%H C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., <a href="http://dx.doi.org/10.1090/S0025-5718-1980-0572872-7">The pseudoprimes to 25*10^9</a>, Mathematics of Computation 35 (1980), pp. 1003-1026.

%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>

%e a(6) = 22178658685 = 5 * 47 * 79 * 1093^2 is a pseudoprime that is not squarefree.

%o (PARI) list(lim)=vecsort(concat(concat(apply(p->select(n->Mod(2, n)^(n-1)==1, p^2*vector(lim\p^2\2, i, 2*i-1)), [1093, 3511])), select(n->Mod(2, n)^n==2, 2*3511^2*vector(lim\3511^2\2, i, i))), , 8) \\ valid up to 4.489 * 10^31, _Charles R Greathouse IV_, Sep 12 2012, changed to include the even terms by _Jianing Song_, Feb 07 2019

%Y Cf. A001567, A013929, A001220, A001262, A219346, A247830, A247831, A295740.

%K nonn

%O 1,1

%A _Rick L. Shepherd_, Mar 16 2009

%E More terms from _Max Alekseyev_, May 09 2010

%E Name changed by _Jianing Song_, Feb 07 2019 to include the even pseudoprimes to base 2 (A006935) as was suggested by _Max Alekseyev_.