A158358 Pseudoprimes to base 2 that are not squarefree, including the even pseudoprimes.

1194649, 12327121, 3914864773, 5654273717, 6523978189, 22178658685, 26092328809, 31310555641, 41747009305, 53053167441, 58706246509, 74795779241, 85667085141, 129816911251, 237865367741, 259621495381, 333967711897, 346157884801, 467032496113, 575310702877, 601401837037, 605767053061

Offset: 1

Author

Rick L. Shepherd, Mar 16 2009

Keywords

• nonn

Comments

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

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]

Examples

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

References

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

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10010 (The even terms inserted by Jianing Song)
• Jan Feitsma, W search: non-squarefree pseudoprimes
• R. G. E. Pinch, The pseudoprimes up to 10^13, Lecture Notes in Computer Science, 1838 (2000), 459-473. - _Felix Fröhlich_, Apr 16 2014
• C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Mathematics of Computation 35 (1980), pp. 1003-1026.
• Index entries for sequences related to pseudoprimes

Crossrefs

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

Programs

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

Extensions

More terms from Max Alekseyev, May 09 2010

Name changed by Jianing Song, Feb 07 2019 to include the even pseudoprimes to base 2 (A006935) as was suggested by Max Alekseyev.