A001220 -id:A001220 - cp's OEIS frontend

A212583 Primes p such that p^2 divides 6^(p-1) - 1.

Original entry on oeis.org

66161, 534851, 3152573

Offset: 1

Felix Fröhlich, May 22 2012

Base 6 Wieferich primes.

Next term > 4.119*10^13. [See Fischer link]

P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1996, page 347

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table p. 5.
François G. Dorais and Dominic Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14.
Richard Fischer, Thema: Fermatquotient B^P-1 == 1 mod (P^2)
Wilfrid Keller and Jörg Richstein, Fermat quotients q_p(a) that are divisible by p.
Eric Weisstein, Fermat Quotient, MathWorld
Wikipedia, Base-a Wieferich primes

Cf. A001220, A014127, A090968, A123692, A123693, A128667, A128668, A128669.

Select[Prime[Range[1000000]], PowerMod[6, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

N=10^9; default(primelimit,N);
forprime(n=2,N,if(Mod(6,n^2)^(n-1)==1,print1(n,", ")));
\\ Joerg Arndt, May 01 2013

A174422 1st Wieferich prime base prime(n).

Original entry on oeis.org

1093, 11, 2, 5, 71, 2, 2, 3, 13, 2, 7, 2, 2, 5

Offset: 1

Jonathan Sondow, Mar 19 2010

Smallest prime p such that p^2 divides prime(n)^(p-1) - 1.
Smallest prime p such that p divides the Fermat quotient q_p((prime(n)) = (prime(n)^(p-1) - 1)/p.
See additional comments, links, and cross-refs in A039951.
a(15) = A039951(47) > 4.1*10^13.

			a(1) = 1093 is the first Wieferich prime A001220. a(2) = 11 is the first Mirimanoff prime A014127.

Wikipedia, Generalized Wieferich primes.
Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (or 0 if unknown)

Cf. A001220, A014127, A039951 = smallest prime p such that p^2 divides n^(p-1) - 1, A125636 = smallest prime p such that prime(n)^2 divides p^(prime(n)-1) - 1.

Cf. A178871 = 2nd Wieferich prime base prime(n).

f[n_] := Block[{b = Prime@ n, p = 2}, While[ PowerMod[b, p - 1, p^2] != 1, p = NextPrime@ p]; p]; Array[f, 14]

forprime(a=2, 20, forprime(p=2, 10^9, if(Mod(a, p^2)^(p-1)==1, print1(p, ", "); next({2}))); print1("--, ")) \\ Felix Fröhlich, Jun 27 2014

a(n) = A039951(prime(n)).

a(n) = 2 if and only if prime(n) == 1 (mod 4). [Jonathan Sondow, Aug 29 2010]

A246568 Near-Wieferich primes (primes p satisfying 2^((p-1)/2) == +-1 + A*p (mod p^2)) with |A| < 10.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 31, 41, 43, 59, 67, 71, 89, 127, 251, 379, 569, 571, 1093, 1427, 1451, 1733, 2633, 2659, 2903, 3511, 13463, 15329, 15823, 26107, 60631, 546097, 2549177, 110057537, 165322639, 209227901, 671499313, 867457663, 3520624567

Offset: 1

Felix Fröhlich, Aug 30 2014

The data section gives all terms up to 10^10. There are eight more terms up to 3*10^15 (see b-file).

A is essentially (A007663(n) modulo A000040(n))/2 (see Crandall et al. (1997), p. 437). The choice of the bound for A is rather arbitrary and selecting a larger A will result in more terms in a specific interval. For any p there exist two values of A whose sum is p, except when p is in A001220, in which case A = 0.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..50 (terms n= 1..49 by Felix Fröhlich)
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp. Vol. 66, Num. 217 (1997), 433-449.
J. Knauer and J. Richstein, The continuing search for Wieferich primes, Math. Comp. Vol. 74, Num. 251 (2005), 1559-1563.
PrimeGrid, Wieferich & near Wieferich Primes p < 11e15
PrimeGrid, WW Statistics

Cf. A001220, A195988, A241014, A244801.

a258367(n) = abs(centerlift(Mod(2, n^2)^((n-1)/2))\/n) \\ after Charles R Greathouse IV in A258367
forprime(p=3, , if(a258367(p) < 10, print1(p, ", "))) \\ Felix Fröhlich, Apr 26 2022

A234810 Primes p such that p^2 divides 14^(p-1) - 1.

Original entry on oeis.org

29, 353, 7596952219

Offset: 1

Felix Fröhlich, Apr 19 2014

Base 14 Wieferich primes.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Petr Ležák, Solutions of equation a^(p-1) mod p^2 = 1, p is prime
P. L. Montgomery, New solutions of a^p-1 == 1 (mod p^2), Math. Comp., 61 (203), 361-363

Cf. A001220.

Select[Prime[Range[200]], Divisible[14^(# - 1) - 1, #^2] &] (* Alonso del Arte, Apr 20 2014 *)
Select[Prime[Range[200]],PowerMod[14,#-1,#^2]==1&] (* The program generates the first two terms of the sequence. To generate the third term, increase the Range constant to 351*10^5, but the program will take a long time to run. *) (* Harvey P. Dale, Jun 19 2021 *)

N=10^9; default(primelimit, N);
forprime(n=2, N, if(Mod(14, n^2)^(n-1)==1, print1(n, ", ")));

A096082 Smallest odd prime p such that p^2 | n^(p-1) - 1.

Original entry on oeis.org

3, 1093, 11, 1093, 20771, 66161, 5, 3, 11, 3, 71, 2693, 863, 29, 29131, 1093, 3, 5, 3, 281

Offset: 1

Lekraj Beedassy, Jul 22 2004

Similar to the sequence A039951 where p=2 is allowed.
a(n^k) <= a(n) for any n,k>1.
a(21) > 1.63*10^14 (see Fischer's link).
For all nonnegative integers n and k, a(n^(n^k)) = a(n). (see puzzle 762 in the links). Also a(n) = 3 if and only if mod(n, 36) is in the set {1, 8, 10, 19, 26, 28, 35}. - Farideh Firoozbakht and Jahangeer Kholdi, Nov 01 2014

C. K. Caldwell, The Prime Glossary, Fermat quotient
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)
W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p
Carlos Rivera, Puzzle 762. Conjecture from Ribenboim's book, The Prime Puzzles and Problems Connection.

Cf. A007663, A001220, A039951, A124121, 124122.

f[n_] := Block[{k = 2}, While[k < 5181800 && PowerMod[n, Prime[k] - 1, Prime[k]^2] != 1, k++ ]; If[k == 5181800, 0, Prime[k]]]; Table[ f[n], {n, 70}] (* Robert G. Wilson v, Jul 23 2004 *)

for(n=2, 20, forprime(p=3, 1e9, if(Mod(n, p^2)^(p-1)==1, print1(p, ", "); next({2}))); print1("--, ")) \\ Felix Fröhlich, Jul 24 2014

a(n) = A039951(n) for all n not of the form 4k+1, while a(4k+1) > A039951(4k+1) = 2. - Alexander Adamchuk, Dec 03 2006

Definition corrected by Alexander Adamchuk, Nov 27 2006
Edited by Max Alekseyev, Oct 07 2009
Edited and updated by Max Alekseyev, Jan 29 2012

A195988 Near-Wieferich primes above 10^9: primes p > 10^9 such that 2^((p-1)/2) == +-1 + A*p (mod p^2) with |A| <= 100, i.e., p=prime(i) such that A258367(i) <= 100.

Original entry on oeis.org

1222336487, 1259662487, 1274153897, 1494408397, 1584392531, 1586651309, 1662410923, 1817972423, 1890830857, 2062661389, 2244893621, 2332252547, 2416644757, 2461090421, 2566816313, 2570948153, 2589186937, 2709711233, 2760945133

Offset: 1

Felix Fröhlich, Sep 26 2011

nonn

There are many near-Wieferich primes below 10^9 (including Wieferich primes 1093 and 3511). However, Crandall, Dilcher and Pomerance searched and reported such primes in the interval [10^9, 4*10^12].

The choice of upper bound for |A| is rather arbitrary.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..149 (terms n = 1..139 from Felix Fröhlich)
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp. Vol. 66, Num. 217 (1997), 433-449.
J. Knauer and J. Richstein, The continuing search for Wieferich primes, Math. Comp. Vol. 74, Num. 251 (2005), 1559-1563.
R. McIntosh, E-Mail to Paul Zimmermann
PrimeGrid, Wieferich & near Wieferich Primes p < 11e15
PrimeGrid, WW Statistics

Cf. A001220, A246568, A258367, A258368.

Edited by Max Alekseyev, Dec 21 2011
New b-file from Felix Fröhlich, Aug 26 2015
Definition amended by Felix Fröhlich, Aug 29 2015

A240719 Numbers k such that 2^k == 1 (mod (k+1)^2).

Original entry on oeis.org

1092, 3510

Offset: 1

Felix Fröhlich, Apr 11 2014

There are only two known terms.
If p is in A001220, then p-1 is in the sequence. If k is in the sequence and k+1 is composite, then any prime factor of k+1 is in A001220 (see fifth comment for a proof). In that case, k+1 could be called a 'Wieferich pseudoprime'.
Any further terms are greater than 1.2 * 10^17. - Charles R Greathouse IV, Apr 12 2014
Both known terms have a periodic binary representation (i.e., 1092 = 010001000100, 3510 = 110110110110), so they are terms of A242139. Also, the ratio between those numbers and their divisor sums is 112/39 in both cases (see Dobson's website in the links and also A239875). Are those facts just coincidences? - Felix Fröhlich, Apr 15 2014
Proof of second part of second comment above: Let q be any odd prime factor of (k+1). Since 2 and q^2 are coprime, it follows from Euler's totient theorem (also known as Euler's theorem or Fermat-Euler theorem) that 2^(phi(q^2)) == 1 (mod q^2). Writing phi(q^2) = q^2 - q = q(q-1), one gets 2^(q(q-1)) == 1 (mod q^2). Taking the q-th root of both sides of the congruence yields 2^(q-1) == 1 (mod q^2). Q.E.D. - Felix Fröhlich, Jun 08 2015
If a(3) exists, it corresponds to A001220(3) - 1, i.e., a(3) + 1 must be prime. This can be shown the following way: Assume that a(3) + 1 is composite. Then the theorem from previous comment implies that a(3) + 1 is of the form 1093^x * 3511^y for some x, y >= 0 and x, y not both 0. If x or y is an integer k > 1, then p = 1093 or p = 3511 satisfies 2^(p-1) == 1 (mod p^(2k)). A quick check with PARI shows that neither 1093 nor 3511 satisfies this congruence for any k > 1. This leaves the case where x = y = 1, which can be excluded as well, since 3837523 is not in A001567. Q.E.D. - Felix Fröhlich, Jun 08 2015

J. Dobes and M. Kures, Search for Wieferich primes through the use of periodic binary strings, Serdica J. Computing, Volume 4, Number 3, 2010, 293-300.
J. B. Dobson, A note on the two known Wieferich primes
W. Johnson, On the nonvanishing of Fermat quotients (mod p), J. reine angew. Math., Issue 292 (Jan 1977), 196-200.
L. C. Washington, On Fermat's last theorem, J. reine angew. Math., Issue 289 (Jan 1977), 115-117.

Cf. A001220, A239875.

fQ[n_] := PowerMod[2, n, (n + 1)^2] == 1; Select[ Range@ 3600, fQ] (* Robert G. Wilson v, Jun 17 2015 *)

isok(n) = lift(Mod(2, (n+1)^2)^n) == 1; \\ Michel Marcus, Apr 12 2014

test(lim)=my(t=1); for(i=0, log(lim)\log(1093), my(n=t); while(n<=lim, if(Mod(2,n^2)^(n-1)==1&&n>1, print(n-1)); n*=3511); t*=1093)
test(1.2e17) \\ Test up to the current search bound for Wieferich primes; Charles R Greathouse IV, Apr 12 2014

A242741 Primes p such that p^2 divides 15^(p-1) - 1.

Original entry on oeis.org

29131, 119327070011

Offset: 1

Felix Fröhlich, May 21 2014

Base 15 Wieferich primes. According to Richard Fischer there is no other term up to approximately 5*10^13.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)

Cf. A001220, A014127, A123692, A212583, A123693, A045616, A111027, A128667, A234810

Select[Prime[Range[1000000]], PowerMod[15, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

forprime(n=2, 10^9, if(Mod(15, n^2)^(n-1)==1, print1(n, ", ")));

A242982 Primes p such that p^2 divides 20^(p-1) - 1.

Original entry on oeis.org

281, 46457, 9377747, 122959073

Offset: 1

Felix Fröhlich, May 28 2014

Base 20 Wieferich primes. According to Richard Fischer, there is no other term up to approximately 5*10^13.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)

Cf. A001220, A014127, A123692, A212583, A123693, A045616, A111027, A039951, A096082.

Select[Prime[Range[1000000]], PowerMod[20, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

forprime(n=2, 10^9, if(Mod(20, n^2)^(n-1)==1, print1(n, ", ")));

A244260 Primes p such that p^2 divides 18^(p-1) - 1.

Original entry on oeis.org

5, 7, 37, 331, 33923, 1284043

Offset: 1

Felix Fröhlich, Jun 24 2014

Base 18 Wieferich primes. According to Richard Fischer there is no other term up to approximately 5*10^13.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)

Cf. A001220, A014127, A123692, A212583, A123693, A045616, A111027, A128667, A234810, A242741, A128668, A090968, A242982, A039951, A096082.

Select[Prime[Range[1000000]], PowerMod[18, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

forprime(n=2, 10^9, if(Mod(18, n^2)^(n-1)==1, print1(n, ", ")));

cp's OEIS Frontend

A212583 Primes p such that p^2 divides 6^(p-1) - 1.

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A174422 1st Wieferich prime base prime(n).

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A246568 Near-Wieferich primes (primes p satisfying 2^((p-1)/2) == +-1 + A*p (mod p^2)) with |A| < 10.

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A234810 Primes p such that p^2 divides 14^(p-1) - 1.

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A096082 Smallest odd prime p such that p^2 | n^(p-1) - 1.

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A195988 Near-Wieferich primes above 10^9: primes p > 10^9 such that 2^((p-1)/2) == +-1 + A*p (mod p^2) with |A| <= 100, i.e., p=prime(i) such that A258367(i) <= 100.

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A240719 Numbers k such that 2^k == 1 (mod (k+1)^2).

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PARI

A242741 Primes p such that p^2 divides 15^(p-1) - 1.

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