A001567 -id:A001567 - cp's OEIS frontend

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

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.

A005939 Pseudoprimes to base 10.

Original entry on oeis.org

9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187, 4521, 5461, 6533, 6541, 6601, 7107, 7471, 7777, 8149, 8401, 8911, 10001, 11111, 11169, 11649, 12403, 12801, 13833, 13981, 14701, 14817, 14911, 15211

Offset: 1

N. J. A. Sloane

nonn

This sequence is a subsequence of A121014 & A121912. In fact the terms are composite terms n of these sequences such that gcd(n,10)=1. Theorem: If both numbers q & 2q-1 are primes(q is in the sequence A005382) and n=q*(2q-1) then 10^(n-1) == 1 (mod n) (n is in the sequence A005939) iff mod(q, 20) is in the set {1, 7, 19}. 91,703,12403,38503,79003,188191,269011,... are such terms. - Farideh Firoozbakht, Sep 15 2006
Composite numbers n such that 10^(n-1) == 1 (mod n). - Michel Lagneau, Feb 18 2012
Composite numbers n such that the number of digits of the period of 1/n divides n-1. - Davide Rotondo, Dec 16 2020

R. K. Guy, Unsolved Problems in Number Theory, A12.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
C. Pomerance & N. J. A. Sloane, Correspondence, 1991
Index entries for sequences related to pseudoprimes

Cf. A001567 (pseudoprimes to base 2), A005382, A121014, A121912.

Select[Range[15300], ! PrimeQ[ # ] && PowerMod[10, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)

A006971 Composite numbers k such that k == +-1 (mod 8) and 2^((k-1)/2) == 1 (mod k).

Original entry on oeis.org

561, 1105, 1729, 1905, 2047, 2465, 4033, 4681, 6601, 8321, 8481, 10585, 12801, 15841, 16705, 18705, 25761, 30121, 33153, 34945, 41041, 42799, 46657, 52633, 62745, 65281, 74665, 75361, 85489, 87249, 90751, 113201, 115921, 126217, 129921, 130561, 149281, 158369

Offset: 1

Richard Pinch

nonn

Previous name was "Terms of A047713 that are congruent to +-1 mod 8".

Complement of (A244626 union A244628) with respect to A047713. - Jianing Song, Sep 18 2018

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A12, pp. 44-50.
Hans Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhäuser, Boston, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..828 from Jianing Song using data from A047713)

Subsequence of A001567 and A047713.

Cf. A244626, A244628, A270697, A270698.

Select[Range[10^5], MemberQ[{1, 7}, Mod[#, 8]] && CompositeQ[#] && PowerMod[2, (# - 1)/2, #] == 1 &] (* Amiram Eldar, Nov 06 2023 *)

This sequence appeared as M5461 in Sloane-Plouffe (1995), but was later mistakenly declared to be an erroneous form of A047713. Thanks to Jianing Song for providing the correct definition. - N. J. A. Sloane, Sep 17 2018

Formal definition by Jianing Song, Sep 18 2018

A176997 Integers k such that 2^(k-1) == 1 (mod k).

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331

Offset: 1

Juri-Stepan Gerasimov, Dec 08 2010

nonn

Old definition was: Odd integers n such that 2^(n-1) == 4^(n-1) == 8^(n-1) == ... == k^(n-1) (mod n), where k = A062383(n). Dividing 2^(n-1) == 4^(n-1) (mod n) by 2^(n-1), we get 1 == 2^(n-1) (mod n), implying the current definition. - Max Alekseyev, Sep 22 2016
The union of {1}, the odd primes, and the Fermat pseudoprimes, i.e., {1} U A065091 U A001567. Also, the union of A006005 and A001567 (conjectured by Alois P. Heinz, Dec 10 2010). - Max Alekseyev, Sep 22 2016
These numbers were called "fermatians" by Shanks (1962). - Amiram Eldar, Apr 21 2024

			5 is in the sequence because 2^(5-1) == 4^(5-1) == 8^(5-1) == 1 (mod 5).

Daniel Shanks, Solved and Unsolved Problems in Number Theory, Spartan Books, Washington D.C., 1962.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

The odd terms of A015919.
Odd integers n such that 2^n == 2^k (mod n): this sequence (k=1), A173572 (k=2), A276967 (k=3), A033984 (k=4), A276968 (k=5), A215610 (k=6), A276969 (k=7), A215611 (k=8), A276970 (k=9), A215612 (k=10), A276971 (k=11), A215613 (k=12).
Cf. A000079, A062173, A062175, A176817.

m = 1; Join[Select[Range[m], Divisible[2^(# - 1) - m, #] &],
Select[Range[m + 1, 10^3], PowerMod[2, # - 1, #] == m &]] (* Robert Price, Oct 12 2018 *)

isok(n) = Mod(2, n)^(n-1) == 1; \\ Michel Marcus, Sep 23 2016

from itertools import count, islice
def A176997_gen(startvalue=1): # generator of terms >= startvalue
    if startvalue <= 1:
        yield 1
    k = 1<<(s:=max(startvalue,1))-1
    for n in count(s):
        if k % n == 1:
            yield n
        k <<= 1
A176997_list = list(islice(A176997_gen(),30)) # Chai Wah Wu, Jun 30 2022

Edited by Max Alekseyev, Sep 22 2016

A214305 Fermat pseudoprimes to base 2 with two prime factors.

Original entry on oeis.org

341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31417, 31609, 31621, 35333, 42799, 49141, 49981, 60701, 60787, 65077, 65281, 80581, 83333, 85489, 88357, 90751, 104653, 123251, 129889, 130561

Offset: 1

Marius Coman, Jul 12 2012

nonn

This sequence is the same as A050217 for the first 60 terms and starts to differ at the 61st.
Conjecture: For any biggest prime factor of a Poulet number p1 with two prime factors, there exists a series with infinitely many Poulet numbers p2 formed this way: p2 mod (p1 - d) = d, where d is the biggest prime factor of p1. Note: it can be seen that the Poulet numbers divisible by 73 bigger than 2701 (7957, 10585, 15841, 31609, etc.) can be written as 1314*n + 73 as well as 2628*m + 73.
Conjecture: Any Poulet number p2 divisible by d can be written as (p1 - d)*n + d, where n is a positive integer, if there exists a smaller Poulet number p1 with two prime factors divisible by d.
Note: This conjecture can't be extrapolated for Poulet numbers p1 with more than two prime factors; for instance, if p1 = 561 = 3*11*17, there indeed are bigger Poulet numbers divisible by 17 (such as 1105 and 4369) that can be written as 544*n + 17, but there also exist such numbers that can't be written this way, e.g., 2465. But the first conjecture can be extrapolated.
Conjecture: For any biggest prime factor of a Poulet number p1 exists a series with infinitely many Poulet numbers p2 formed this way: p2 mod (p1 - d) = d, where d is the biggest prime factor of p1.
For each prime p, there are only a finite number of q such that p*q is here. See A085014. Sequence A180471 lists the factors of terms of this sequence. - T. D. Noe, Sep 20 2012
Numbers n = p*q such that n divides 2^(p-1)-1 and 2^(q-1)-1, where p,q are primes; thus 2^gcd(p-1,q-1) == 1 (mod n). - Thomas Ordowski, Aug 27 2016
These are semiprimes p*q such that 2^(p+q-2) == 1 (mod p*q). Proof: 2^(p-1) == 1 (mod p) and 2^(q-1) == 1 (mod q), so 2^((p-1)*(q-1)) == 1 (mod p*q), and (p-1)*(q-1) = (p*q-1)-(p+q-2). - Amiram Eldar and Thomas Ordowski, Apr 02 2021

			Few examples for the first 4 Poulet numbers with two prime factors:
For p1 = 341 = 11*31, the following Poulet numbers p2 for which p2 mod 310 = 31 were obtained: 2821, 4371, 4681, 10261 etc.
For p1 = 1387 = 19*73, the following Poulet numbers p2 for which p2 mod 1314 = 73 were obtained: 2701, 7957, 10585, 15841 etc.
For p1 = 2047 = 23*89, the following Poulet numbers p2 for which p2 mod 1958 = 89 were obtained: 31417, 35333, 60787, 62745 etc.
For p1 = 2701 = 37*73, the following Poulet numbers p2 for which p2 mod 2628 = 73 were obtained: 7957, 10585, 15841 etc.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 3299 terms from T. D. Noe)
Marius Coman, Conjecture that states that a Mersenne number with odd exponent is either prime either divisible by a 2-Poulet number, 2015.
Eric Weisstein's World of Mathematics, Poulet Number.
Index entries for sequences related to pseudoprimes

Subsequence of A050217.

Cf. A001567.

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; Select[Range[200000], SemiPrimeQ[#] && PowerMod[2, #-1, #] == 1 &] (* T. D. Noe, Jul 12 2012 *)

list(lim)=my(v=List());forprime(p=31,lim\11, forprime(q=11,min(p-1,lim\p), if(Mod(2,p)^(q-1)==1 && Mod(2,q)^(p-1)==1, listput(v,p*q)))); if(lim>=1093^2,listput(v,1093^2)); if(lim>=3511^2,listput(v,3511^2)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 20 2012

A005937 Pseudoprimes to base 6.

Original entry on oeis.org

35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729, 2465, 2701, 2821, 3421, 3565, 3589, 3913, 4123, 4495, 5713, 6533, 6601, 8029, 8365, 8911, 9331, 9881, 10585, 10621, 11041, 11137, 12209, 14315, 14701, 15841, 16589, 17329, 18361, 18721, 20017, 21049, 22049

Offset: 1

N. J. A. Sloane

nonn

Theorem: If both numbers q and 2q-1 are primes and n=q*(2q-1) then 6^(n-1) == 1 (mod n) (n is in the sequence) iff q is of the form 12k+1. 2701, 18721, 49141, 104653, 226801, 665281, ... are such terms. This sequence is a subsequence of A122783. - Farideh Firoozbakht, Sep 12 2006

Composite numbers k such that 6^(k-1) == 1 (mod k). - Michel Lagneau, Feb 18 2012

R. K. Guy, Unsolved Problems in Number Theory, A12.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, T. D. Noe, and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (Mathar 1..118, Noe 119..1000, Greathouse 1001..10000)
C. Pomerance & N. J. A. Sloane, Correspondence, 1991
Index entries for sequences related to pseudoprimes

Cf. A001567 (pseudoprimes to base 2), A122783.

Select[Range[20000], ! PrimeQ[ # ] && PowerMod[6, #-1, # ] == 1 &] (* Farideh Firoozbakht, Sep 12 2006 *)

More terms from Farideh Firoozbakht, Sep 12 2006

A083876 Least pseudoprime to base 2 through base prime(n).

Original entry on oeis.org

341, 1105, 1729, 29341, 29341, 162401, 252601, 252601, 252601, 252601, 252601, 252601, 1152271, 2508013, 2508013, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 6733693, 6733693, 6733693

Offset: 1

Robert G. Wilson v, May 06 2003

nonn

Records: 341, 1105, 1729, 29341, 162401, 252601, 1152271, 2508013, 3828001, 6733693, 17098369, 17236801, 29111881, 82929001, 172947529, 216821881, 228842209, 366652201, .... - Robert G. Wilson v, May 11 2012
Conjecture: for n > 1, a(n) is the smallest Carmichael number k with lpf(k) > prime(n). It seems that such Carmichael numbers have exactly three prime factors. - Thomas Ordowski, Apr 18 2017
The conjecture is true if a(n) < A285549(n) for all n > 1. It holds for all a(n) < 2^64. - Max Alekseyev and Thomas Ordowski, Mar 13 2018
If prime(n) < m < a(n), then m is prime if and only if p^(m-1) == 1 (mod m) for every prime p <= prime(n). - Thomas Ordowski, Mar 05 2018
By this conjecture in the second comment, a(n) <= A135720(n+1), with equality for n > 1 iff a(n) < a(n+1), namely for n = 2, 3, 5, 6, 12, 13, 15, 25, 28, 29, ... For such n, a(n) gives all terms of A300629 > 561. - Thomas Ordowski, Mar 10 2018

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 100 terms from Robert G. Wilson v)

Cf. A002997, A001567, A052155, A083737, A087788, A083739, A135720, A141768, A250199, A271221, A285549, A300629.

k = 4; Do[l = Table[ Prime[i], {i, 1, n}]; While[ PrimeQ[k] || Union[PowerMod[l, k - 1, k]] != {1}, k++ ]; Print[k], {n, 1, 29}]

isps(k, n) = {if (isprime(k), return (0)); my(nbok = 0); for (b=2, prime(n), if (Mod(b, k)^(k-1) == 1, nbok++, break)); if (nbok==prime(n)-1, return (1));}
a(n) = {my(k=2); while (!isps(k, n), k++); return (k);} \\ Michel Marcus, Apr 27 2018

A016035 a(n) = Sum_{j|n, 1 < j < n} phi(j). Also a(n) = n - phi(n) - 1 for n > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 3, 2, 5, 0, 7, 0, 7, 6, 7, 0, 11, 0, 11, 8, 11, 0, 15, 4, 13, 8, 15, 0, 21, 0, 15, 12, 17, 10, 23, 0, 19, 14, 23, 0, 29, 0, 23, 20, 23, 0, 31, 6, 29, 18, 27, 0, 35, 14, 31, 20, 29, 0, 43, 0, 31, 26, 31, 16, 45, 0, 35, 24, 45, 0, 47, 0, 37, 34, 39, 16, 53

Offset: 1

Robert G. Wilson v

Number of integers less than n with at least one common factor with n. - Olivier Gérard, Feb 08 2011
A number N is a Fermat base 2 pseudoprime, that is, 2^(N-1) == 1 mod N, iff 2^a(N) == 1 mod N. - T. D. Noe, Jul 10 2003
Number of zero divisors in ring Z_n, where Z_n is the ring of integers modulo n. - Armin Vollmer (armin_vollmer(AT)web.de), Jul 23 2004
From Jianing Song, Apr 20 2019: (Start)
a(p) = 0 if and only if p is a prime, which is equivalent to the fact that Z_p is a field if and only if p is a prime.
a(n) = n/2 is and only if n = 2p, p prime. (End)

			For n = 6, the a(6) = 3 integers less than 6 with at least one common factor with 6 are {2,3,4}.

Al Hibbard and Ken Levasseur, "Exploring Abstract Algebra with Mathematica", Springer Verlag.

Olivier Gérard, Table of n, a(n) for n = 1..10000

Cf. A001567 (base 2 pseudoprimes).
Cf. A000010, A027750.
Essentially one less than cototient, A051953.

a016035 1 = 0
a016035 n = sum $ map a000010 $ init $ tail $ a027750_row n
-- Reinhard Zumkeller, Mar 02 2012

Needs["AbstractAlgebra`Master`"] Length[ZeroDivisors[Z[ # ]]] & /@ Range[2, 25] (* Armin Vollmer, Jul 23 2004 *)
a[n_] := n - EulerPhi[n] - 1; a[1] = 0; Table[a[n], {n, 1, 78}] (* Jean-François Alcover, Jan 04 2013 *)

for(n=1,100,p=0;for(i=1,n-1,if(gcd(i,n)>1,p++));print1(p",")) /* V. Raman, Nov 22 2012 */

for(n=1,100,if(n==1,print1(0","),print1(n-1-eulerphi(n)","))) /* V. Raman, Nov 22 2012 */

For n > 1, a(n) = A051953(n) - 1. - Antti Karttunen, Mar 12 2018

Typo in definition fixed by Reinhard Zumkeller, Mar 02 2012

A020138 Pseudoprimes to base 9.

Original entry on oeis.org

4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288, 1387, 1541, 1729, 1891, 2465, 2501, 2665, 2701, 2806, 2821, 2926, 3052, 3281, 3367, 3751, 4376, 4636, 4961, 5356, 5551, 6364, 6601, 6643, 7081, 7381, 7913, 8401

Offset: 1

David W. Wilson

nonn

This sequence is a subsequence of A122786. In fact the terms are composite terms n of A122786 such that gcd(n,3)=1. Theorem: If both numbers q & 2q-1 are primes greater than 3 and n=q*(2q-1) then 9^(n-1)==1 (mod n) (n is in the sequence). So for n>2 A005382(n)* (2*A005382(n)-1) is in the sequence; 91,703,1891,2701,12403,18721,... is the related subsequence. - Farideh Firoozbakht, Sep 15 2006

Composite numbers n such that 9^(n-1) == 1 (mod n).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..159 from R. J. Mathar, terms 160..1000 from T. D. Noe)
Index entries for sequences related to pseudoprimes

Cf. A001567 (pseudoprimes to base 2), A005382, A122786.

Select[Range[8500], ! PrimeQ[ # ] && PowerMod[9, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)

A057942 Pseudoprimes k to base 2 such that k-2 and k+2 are primes.

Original entry on oeis.org

645, 25761, 212421, 332949, 656601, 6212361, 63560685, 413435121, 1112691009, 1121315385, 1239702465, 2851612221, 5616154545, 6454103601, 6689540901, 9691152801, 9728582781, 11970358401, 12028438605, 15076352901, 17402551365

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

B-file extended using Feitsma's tables of pseudoprimes. - Giovanni Resta, Aug 20 2018

Giovanni Resta, Table of n, a(n) for n = 1..7000 (first 73 terms from Amiram Eldar)
C. K. Caldwell, Pseudoprimes
Jan Feitsma and William Galway, Tables of pseudoprimes and related data
Andrzej Rotkiewicz, On pseudoprimes having special forms and a solution of K. Szymiczek's problem, Acta Mathematica Universitatis Ostraviensis, Vol. 13, No. 1 (2005), pp. 57-71.
Index entries for sequences related to pseudoprimes

Cf. A001567.

More terms from Don Reble, Nov 03 2001

cp's OEIS Frontend

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

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A005939 Pseudoprimes to base 10.

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A006971 Composite numbers k such that k == +-1 (mod 8) and 2^((k-1)/2) == 1 (mod k).

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A176997 Integers k such that 2^(k-1) == 1 (mod k).

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A214305 Fermat pseudoprimes to base 2 with two prime factors.

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A005937 Pseudoprimes to base 6.

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A083876 Least pseudoprime to base 2 through base prime(n).

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