A014127 -id:A014127 - 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]

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, ", ")));

A242958 Numbers n such that 3^phi(n) == 1 (mod n^2), where phi(n) is Euler's totient function.

Original entry on oeis.org

11, 22, 44, 55, 110, 220, 440, 880, 1006003, 2012006, 4024012, 11066033, 22132066, 44264132, 55330165, 88528264, 110660330, 221320660, 442641320, 885282640, 1770565280, 56224501667, 112449003334, 224898006668, 393571511669, 449796013336, 618469518337, 787143023338

Offset: 1

Felix Fröhlich, May 27 2014

a(21) > 10^9.

All listed composite terms are multiples of the two known primes in this sequence, 11 and 1006003, the only known base 3 Wieferich primes.

Giovanni Resta, Table of n, a(n) for n = 1..78 (terms < 10^15)

Cf. A014127, A077816, A242959, A242960, A000010.

Select[Range[1000], Mod[3^EulerPhi[#], #^2] == 1 &] (* Alonso del Arte, Jun 02 2014 *)

for(n=2, 10^9, if(Mod(3, n^2)^(eulerphi(n))==1, print1(n, ", ")))

Terms a(21) and beyond from Giovanni Resta, Jan 27 2020

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, ", ")));

A244065 Pseudoprimes to base 3 that are not squarefree.

Original entry on oeis.org

121, 3751, 4961, 7381, 11011, 29161, 32791, 142901, 228811, 239701, 341341, 551881, 566401, 595441, 671671, 784201, 856801, 1016521, 1237951, 1335961, 1433971, 1804231

Offset: 1

Felix Fröhlich, Jun 19 2014

nonn

Must be divisible by the square of a Mirimanoff prime, A014127. - Charles R Greathouse IV, Jun 21 2014

Felix Fröhlich and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 196 terms from Fröhlich)

Cf. A005935, A158358, A014127.

for(n=2, 10^9, if(!isprime(n) && Mod(3, n)^(n-1)==1 && !issquarefree(n), print1(n, ", ")))

list(lim)=my(M=[11,1006003],v=List(),p2);for(i=1,#M,p2=M[i]^2;forstep(n=p2,lim,p2,if(Mod(3,n)^(n-1)==1,listput(v,n))));Set(v) \\ Good for lim <= 9.4 * 10^29; Charles R Greathouse IV, Jun 21 2014

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, ", ")));

A178871 2nd Wieferich prime base prime(n).

Original entry on oeis.org

3511, 1006003, 20771, 491531

Offset: 1

Jonathan Sondow, Jun 20 2010, Jun 24 2010

2nd prime p such that p^2 divides prime(n)^(p-1) - 1.
2nd prime p such that p divides the Fermat quotient q_p(p_n) = ((p_n)^(p-1) - 1)/p, where p_n = prime(n).
a(5) is unknown: 71 is the only known prime p that divides q_p(11).
If a(5) is found, the sequence continues a(6) = 863, a(7) = 3, a(8) = 7, a(9) = 2481757.
See additional comments, references, links, and cross-refs in A039951 and A174422.

			a(1) = 3511 is the 2nd Wieferich prime A001220(2).
a(2) = 1006003 is the 2nd Mirimanoff prime A014127(2).

Wikipedia, Generalized Wieferich primes

Cf. A039951 = smallest prime p such that p^2 divides n^(p-1) - 1, A174422 = first Wieferich prime base prime(n).

{default(primelimit, 10^7); for(n=1, 9, a=prime(n); c=0; forprime(p=2, 10^7, if(Mod(a, p^2)^(p-1)==1, c++; if(c==2, print1(p, ", "); next(2)))); print1(">10^7, "))} \\ Jens Kruse Andersen, Jun 18 2014

A109641 Composite n such that binomial(3n, n) == 3^k (mod n) for some integer k > 0.

Original entry on oeis.org

4, 9, 15, 25, 27, 34, 36, 49, 51, 57, 63, 68, 75, 81, 87, 93, 111, 121, 125, 129, 132, 138, 141, 153, 155, 159, 169, 177, 237, 249, 258, 261, 264, 267, 274, 276, 279, 289, 298, 303, 324, 339, 343, 357, 361, 375, 381, 387, 393, 411, 417, 423, 441, 447, 453, 477

Offset: 1

Ryan Propper, Aug 05 2005

nonn

Includes p^k for k >= 2 and p > 2 in A019334 but not in A014127, as binomial(3n,n) is coprime to p and 3 is a primitive root mod p^k. - Robert Israel, Nov 12 2017

			Binomial(3*34,34) == 3^6 (mod 34), so 34 is a member.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A019334, A080469, A014127, A109642.

filter:= proc(n) local p,m,k,t;
  if isprime(n) then return false fi;
  p:= padic:-ordp(n,3);
  p:= p + numtheory:-order(3, n/3^p);
  m:= binomial(3*n,n) mod n;
  t:= 1;
  for k from 1 to p do
    t:= t*3 mod n;
    if t = m then return true fi;
  od:
false
end proc;
select(filter, [$2..1000]); # Robert Israel, Nov 12 2017

okQ[n_] := Module[{p, m}, If[PrimeQ[n], Return[False]]; p = IntegerExponent[n, 3]; p = p + MultiplicativeOrder[3, n/3^p]; m = Mod[Binomial[3n, n], n]; AnyTrue[Range[p], m == PowerMod[3, #, n]&]];
Select[Range[2, 500], okQ] (* Jean-François Alcover, Mar 27 2019, after Robert Israel *)

Corrected and extended by Max Alekseyev, Sep 13 2009

Edited by Max Alekseyev, Sep 20 2009

A125774 Numbers k such that 3^k mod k = 3^k mod k^2.

Original entry on oeis.org

1, 2, 3, 4, 9, 11, 20, 22, 27, 33, 81, 99, 220, 243, 644, 729, 1220, 2187, 2420, 5060, 6561, 7128, 8368, 13420, 14740, 19683, 23620, 40573, 55660, 59049, 145420, 147620, 162140, 177147, 237820, 259820, 290620, 308660, 339020, 447740, 531441, 548660

Offset: 1

Alexander Adamchuk, Dec 07 2006

nonn

This sequence includes all powers of 3. a(2) = 2, a(3) = 3, a(6) = 11 and a(45) = 1006003 are the only known primes in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..400 (terms 1..100 from Harvey P. Dale)

Cf. A014127 (Primes p such that p^2 divides 3^(p-1) - 1).
Cf. A068535 (Numbers k such that 2^k mod k = 2^k mod k^2).
Cf. A125773 (Numbers k, that are not powers of 2, such that 2^k mod k = 2^k mod k^2).
Cf. A125775 (Numbers k such that 5^k mod k = 5^k mod k^2).

Do[f=PowerMod[3,n,n];g=PowerMod[3,n,n^2];If[f==g,Print[n]],{n,1,1100000}]
Select[Range[600000],PowerMod[3,#,#]==PowerMod[3,#,#^2]&] (* Harvey P. Dale, Feb 21 2013 *)

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

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A242958 Numbers n such that 3^phi(n) == 1 (mod n^2), where phi(n) is Euler's totient function.

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

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A244065 Pseudoprimes to base 3 that are not squarefree.

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

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A178871 2nd Wieferich prime base prime(n).

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