A295740 -id:A295740 - cp's OEIS frontend

A006935 Even pseudoprimes (or primes) to base 2: even n that divide 2^n - 2.

2, 161038, 215326, 2568226, 3020626, 7866046, 9115426, 49699666, 143742226, 161292286, 196116194, 209665666, 213388066, 293974066, 336408382, 377994926, 410857426, 665387746, 667363522, 672655726, 760569694, 1066079026, 1105826338, 1423998226, 1451887438, 1610063326, 2001038066, 2138882626, 2952654706, 3220041826

Offset: 1

N. J. A. Sloane, Richard C. Schroeppel

Of course, 2 is the only true prime here.
Numbers a(n)/2 form the odd terms of A130421. - Max Alekseyev, May 28 2014
a(n) == 2 (mod 4), hence there are no consecutive even numbers in this sequence. The closest two terms below 2*10^15 (as computed by Alekseyev) are a(2) = 161038 and a(3) = 215326 with a(3) - a(2) = 54288. Do smaller gaps exist? - Charles R Greathouse IV, Dec 02 2014
Corollary (Rotkiewicz-Ziemak, 1995): 2(2^p-1)(2^q-1) is a pseudoprime if and only if 2(2^(pq)-1) is a pseudoprime, where p,q are distinct primes. - Thomas Ordowski, Apr 09 2016
Numbers 2k such that 2^(2k-1) == 1 (mod k). - Thomas Ordowski, Nov 22 2016
There exist even pseudoprimes that are not squarefree, with the smallest being 190213279479817426 = 2 * 7 * 79 * 1951 * 3511^2 * 7151 (cf. A295740). - Max Alekseyev, Nov 26 2017
Terms of the form 2^k - 2 corresponds to k in A296104. - Max Alekseyev, Dec 04 2017
From Bernard Schott, Oct 11 2021: (Start)
Two significant dates in the history of these terms:
1949: Derrick Henry Lehmer finds the smallest even pseudoprime to base 2, a(2) = 161038 (see Lehmer link).
1951: Dutch mathematician N. G. W. H. Beeger proves that the number of even pseudoprimes is infinite (see Beeger link). (End)

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 23.
J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of b^n+/-1 b=2, 3, 5, 6, 7, 10, 11, 12 up to high powers, Contemporary Math. v.22.
R. K. Guy, Unsolved Problems in Number Theory, A12.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 91.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of n, a(n) for n = 1..1319 (contains all terms below 2*10^15; first 156 terms from R. G. E. Pinch)
N. G. W. H. Beeger, On even numbers m dividing 2^m-2, Amer. Math. Monthly, Vol. 58, No. 8 (1951), pp. 553-555.
D. H. Lehmer, On the Converse of Fermat's Theorem II, Amer. Math. Monthly, Vol. 56, No. 5 (1949), pp. 300-309.
A. Rotkiewicz and K. Ziemak, On Even Pseudoprimes, The Fibonacci Quarterly, Vol. 33, No. 2 (1995), pp. 123-125.
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Index entries for sequences related to pseudoprimes

The even terms of A015919.

Cf. A295740.

Select[2*Range[5000000],PowerMod[2,#,#]==2&] (* Harvey P. Dale, Dec 02 2012 *)

is(n)=Mod(2,n)^n==2 && n%2==0 \\ Charles R Greathouse IV, Dec 02 2014

More terms from Robert G. Wilson v
Corrected by T. D. Noe, May 27 2003
b-file corrected by Max Alekseyev, Oct 09 2016

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

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Mar 16 2009

nonn

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]

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

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

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

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

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

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.

A247831 Pseudoprimes to base 2 divisible by 3511^2, including the even pseudoprimes.

Original entry on oeis.org

12327121, 129816911251, 259621495381, 346157884801, 605767053061, 6317168754781, 6922923480721, 12634325182441, 18518799663001, 21634109682121, 24273469559431, 57114029344321, 65681131896901, 102718706568661, 135083316211741, 135818875521811, 153342494379361

Offset: 1

Felix Fröhlich, Sep 24 2014

nonn

Numbers k such that 2^k == 2 (mod k) and k is divisible by 3511^2.

Unless there are other Wieferich primes (A001220) besides 1093 and 3511, the intersection and the union of this sequence with A247830 are given by A219346 and A158358, respectively, and the even terms are given by A295740. - Max Alekseyev, Nov 26 2017 [The indices of the even terms in this sequence are 430, 525, 543, 701, 811, 826, 937, 1235, 1277, 1388, ... - Jianing Song, Feb 08 2019]

Jianing Song, Table of n, a(n) for n = 1..1470
R. G. E. Pinch, The pseudoprimes up to 10^13, Lecture Notes in Computer Science, 1838 (2000), 459-473.
C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Mathematics of Computation 35 (1980), pp. 1003-1026.

Subsequence of each of (A001567 U A006935), A015919, A158358 composed of the terms divisible by 3511^2.

Cf. A001220, A013929, A219346, A247830, A295740.

vi=readvec("b158358.txt")
for(n=1, #vi, if(Mod(vi[n], 3511^2)==0, print1(vi[n], ", ")))

list(N)=select(k->Mod(2, k)^k==2, 3511^2*vector(N\3511^2\2, i, i)) \\ Jianing Song, Feb 07 2019

Name changed by Jianing Song, Feb 07 2019 to include the even pseudoprimes to base 2 (A006935) at the suggestion of Max Alekseyev.

A306450 Non-coprime pseudoprimes to base 3 (A306451) that are not squarefree.

Original entry on oeis.org

726, 1053426, 6498426, 7912311, 8141001, 190381521, 202730781, 283975626, 524245326, 767159481, 1095790641, 1620456321, 1904467521, 2287621281, 2700546486, 3462782961, 4120800321, 4928482581, 5816852481, 5974336401, 9313587921, 18723332001, 21215225361, 22073079666, 29882080866, 30132305841

Offset: 1

Jianing Song, Feb 16 2019

nonn

Intersection of A306451 and A013929.

Terms must be divisible by the square of a Mirimanoff prime p (or base-3 Wieferich prime, A014127) such that the multiplicative order of 3 modulo p is not divisible by 3. So far, the only known Mirimanoff primes are 11 and 1006003. The multiplicative order of 3 modulo 11 is 5, not a multiple of 3, while the multiplicative order of 3 modulo 1006003 is 1006002, which is a multiple of 3. As a result, all known terms are divisible by 3*11^2 = 363.

			726 is a term because 726 divides 3^726 - 3 and 726 = 2 * 3 * 11^2.

Cf. A295740, A244065, A306451, A306452.

forstep(n=3, 10^9, 3, if(Mod(3, n)^n==3 && !issquarefree(n), print1(n, ", ")))

More terms from Jinyuan Wang, Feb 18 2019

A353221 Squarefree base-2 Fermat pseudoprimes divisible by a Wieferich prime.

Original entry on oeis.org

3581761, 5173169, 5968873, 23872213, 36974341, 53711113, 107023281, 427294141, 490950461, 526359289, 546649741, 550230409, 753532781, 803264281, 836683849, 1115578101, 1168492417, 1193557093, 1540454761, 1632785701, 2129304997, 2295209281, 2677147201

Offset: 1

Felix Fröhlich, May 01 2022

nonn

Subsequence of A001567.

The least term that is divisible by both 1093 and 3511 is a(799) = 7015325908501 = 937 * 1093 * 1951 * 3511. - Amiram Eldar, May 05 2022

			3581761 = 29 * 113 * 1093, so it is a base-2 pseudoprime divisible by the Wieferich prime 1093 and is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
PrimeGrid, Subproject status.

Cf. A001220, A001567, A158358, A219346, A247830, A247831, A295740.

/* The following program is valid up to the current search bound for Wieferich primes, about 10^19 as of May 03 2022 (cf. PrimeGrid); the program may miss terms above that bound if there is another Wieferich prime */
forcomposite(c=1, , if(Mod(2, c)^(c-1)==1, if(Mod(c, 1093)==0 || Mod(c, 3511)==0, if(issquarefree(c), print1(c, ", ")))))

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A006935 Even pseudoprimes (or primes) to base 2: even n that divide 2^n - 2.

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

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A247831 Pseudoprimes to base 2 divisible by 3511^2, including the even pseudoprimes.

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A306450 Non-coprime pseudoprimes to base 3 (A306451) that are not squarefree.

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A353221 Squarefree base-2 Fermat pseudoprimes divisible by a Wieferich prime.

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