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A158358
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Pseudoprimes to base 2 that are not squarefree, including the even pseudoprimes.
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13
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1194649, 12327121, 3914864773, 5654273717, 6523978189, 22178658685, 26092328809, 31310555641, 41747009305, 53053167441, 58706246509, 74795779241, 85667085141, 129816911251, 237865367741, 259621495381, 333967711897, 346157884801, 467032496113, 575310702877, 601401837037, 605767053061
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OFFSET
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1,1
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COMMENTS
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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.
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
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
Unless there are other Wieferich primes besides 1093 and 3511, the sequence is the union of A247830 and A247831. - Max Alekseyev, Nov 26 2017
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]
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REFERENCES
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P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, NY, 1991, pp. 77, 83, 167.
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LINKS
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C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Mathematics of Computation 35 (1980), pp. 1003-1026.
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EXAMPLE
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a(6) = 22178658685 = 5 * 47 * 79 * 1093^2 is a pseudoprime that is not squarefree.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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