A001220 -id:A001220 - cp's OEIS frontend

A123692 Primes p such that p^2 divides 5^(p-1) - 1.

2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801

Offset: 1

Max Alekseyev, Oct 07 2006

Dorais and Klyve proved that there are no further terms up to 9.7*10^14.
From Felix Fröhlich, Jan 06 2017: (Start)
a(6) and a(7) were found by Keller and Richstein (cf. Keller, Richstein, 2005).
Prime terms of A242959. (End)
From Jianing Song, Jun 21 2025: (Start)
The ring of integers of Q(5^(1/k)) is Z[5^(1/k)] if and only if k does not have a prime factor in this sequence (k is even or in A342391). See Theorem 5.3 of the paper of Keith Conrad. For example, we have:
  (1 + sqrt(5))/2 is an algebraic integer, but it is not in Z[sqrt(5)];
  (1 + 5^(10385/20771) + 5^(2*10385/20771) + ... + 5^(10384*10385/20771))/20771 is an algebraic integer, but it is not in Z[5^(1/20771)];
  (1 + 5^(40486/40487) + 5^(2*40486/40487) + ... + 5^(40486*40486/40487))/40487 is an algebraic integer, but it is not in Z[5^(1/40487)]. (End)

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Chris K. Caldwell, The Prime Glossary, Fermat quotient.
Keith Conrad, The ring of integers in a radical extension.
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.
W. Keller and J. Richstein, Solutions of the congruence a^p-1 == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
A. Paszkiewicz, A new prime p for which the least primitive root (mod p) and the least primitive root (mod p^2) are not equal, Math. Comp. 78 (2009), 1193-1195.

Cf. A001220, A014127, A123693, A128667, A128668, A090968, A128669, A096082, A242959.

Select[Prime[Range[2500]], Divisible[5^(# - 1) - 1, #^2] &] (* Alonso del Arte, Aug 01 2014 *)
Select[Prime[Range[55*10^6]],PowerMod[5,#-1,#^2]==1&] (* The program generates the first 4 terms of the sequence. *) (* Harvey P. Dale, Jan 29 2023 *)

N=10^9; default(primelimit, N);
forprime(n=2, N, if(Mod(5, n^2)^(n-1)==1, print1(n, ", ")));
\\ Joerg Arndt, May 01 2013

More terms from Alexander Adamchuk, Nov 27 2006

Updated by Max Alekseyev, Jan 29 2012

A141232 Overpseudoprimes to base 2: composite k such that k = A137576((k-1)/2).

Original entry on oeis.org

2047, 3277, 4033, 8321, 65281, 80581, 85489, 88357, 104653, 130561, 220729, 253241, 256999, 280601, 390937, 458989, 486737, 514447, 580337, 818201, 838861, 877099, 916327, 976873, 1016801, 1082401, 1145257, 1194649, 1207361, 1251949, 1252697, 1325843

Offset: 1

Vladimir Shevelev, Jun 16 2008

nonn

Numbers are found by prime factorization of numbers from A001262 and a simple testing of the conditions indicated in comment to A141216.
All composite Mersenne numbers (A001348), Fermat numbers (A000215) and squares of Wieferich primes (A001220) are in this sequence. - Vladimir Shevelev, Jul 15 2008
C. Pomerance proved that this sequence is infinite (see Theorem 4 in the third reference). - Vladimir Shevelev, Oct 29 2011
Odd composite numbers k such that ord(2,k) * ((Sum_{d|k} phi(d)/ord(2,d)) - 1) = k-1, where phi = A000010 and ord(2,d) is the multiplicative order of 2 modulo d. - Jianing Song, Nov 13 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..664 from Michel Marcus)
J. H. Castillo, G. García-Pulgarín and J. M. Velásquez-Soto, q-pseudoprimality: A natural generalization of strong pseudoprimality, arXiv:1412.5226 [math.NT], 2014.
Vladimir Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich primes, arXiv:0806.3412 [math.NT], 2008-2012.
Vladimir Shevelev, Process of "primoverization" of numbers of the form a^n-1, arXiv:0807.2332 [math.NT], 2008.
Vladimir Shevelev, On upper estimate for the overpseudoprime counting function, arXiv:0807.1975 [math.NT], 2008.
Vladimir Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers, J. Integer Seq. 15 (2012) Article 12.7.7.
Vladimir Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers, arXiv:1206.0606 [math.NT], 2012.

Cf. A000010, A001262, A141216, A001567, A002326.

A137576[n_] := Module[{t}, (t = MultiplicativeOrder[2, 2 n + 1])* DivisorSum[2 n + 1, EulerPhi[#]/MultiplicativeOrder[2, #]&] - t + 1];
okQ[n_] := n > 1 && CompositeQ[n] && n == A137576[(n-1)/2];
Reap[For[k = 2, k < 2*10^6, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jan 11 2019, from PARI *)

f(n)=my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
isok(n) = (n>1) && !isprime(n) && (n == f((n-1)/2)); \\ Michel Marcus, Oct 05 2018

Sum_{n:a(n)<=x} 1 <= x^(3/4)(1+o(1)).

Name edited by Michel Marcus, Oct 05 2018

A123693 Primes p such that p^2 divides 7^(p-1) - 1.

Original entry on oeis.org

5, 491531

Offset: 1

Max Alekseyev, Oct 07 2006

Dorais and Klyve proved that there are no further terms up to 9.7*10^14.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
F. G. Dorais and D. Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14.
W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p.

Cf. A001220, A014127, A123692, A123693, A128667, A128668, A090968, A039951, A128669

Select[Prime[Range[1000000]], PowerMod[7, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

Updated by Max Alekseyev, Jan 29 2012

A077816 Wieferich numbers (1): n > 1 such that 2^A000010(n) == 1 (mod n^2).

Original entry on oeis.org

1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995, 228215, 298389, 410787, 473985, 684645, 895167, 1232361, 2053935, 2685501, 3697083, 3837523, 6161805, 11512569

Offset: 1

Reinhard Zumkeller, Nov 17 2002

nonn

A077815(a(n)) = 1.
The only known primes are a(1)=A001220(1)=1093 and a(3)=A001220(2)=3511, the Wieferich primes.
If there are finitely many Wieferich primes (A001220), this sequence is finite. In particular, unless there are other Wieferich primes besides 1093 and 3511, this sequence consists of 104 terms with the largest being 16547533489305 (Agoh et al., 1997).
a(105)=A001220(3) in the sense that either both numbers are well-defined and equal, or else neither number exists. - Jeppe Stig Nielsen, Oct 16 2016

			A077815(3279) = 2^A000010(3279) mod 3279^2 = 2^2184 mod 10751841 = 1, therefore 3279 is a term.

Max Alekseyev, Table of n, a(n) for n = 1..104 (all currently known terms)
T. Agoh, K. Dilcher, and L. Skula, Fermat Quotients for Composite Moduli, Journal of Number Theory 66(1), 1997, 29-50. doi: 10.1006/jnth.1997.2162
William D. Banks, Florian Luca, and Igor E. Shparlinski, Estimates for Wieferich Numbers, The Ramanujan Journal, December 2007, Volume 14, Issue 3, pp 361-378.
R. Crandall, K. Dilcher, and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, Volume 66, 1997.
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001, p. 28.
Jiří Klaška, A Simple Proof of Skula's Theorem on Prime Power Divisors of Mersenne Numbers, J. Int. Seq., Vol. 25 (2022), Article 22.4.3.
Jiří Klaška, Jakóbczyk's Hypothesis on Mersenne Numbers and Generalizations of Skula's Theorem, J. Int. Seq., Vol. 26 (2023), Article 23.3.8.

For another definition of Wieferich numbers, see A182297.

Cf. A001220.

[n: n in [1..8*10^5] | 2^EulerPhi(n) mod n^2 eq 1]; // Vincenzo Librandi, Dec 05 2015

Reap[For[k = 1, k <= 10^8, k++, If[PowerMod[2, EulerPhi[k], k^2] == 1, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Nov 17 2021 *)

for(n=2, 10^9, if(Mod(2, n^2)^(eulerphi(n))==1, print1(n, ", "))); \\ Felix Fröhlich, May 27 2014

More terms from Emeric Deutsch, Mar 05 2005

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Jun 18 2005

A128667 Primes p such that p^2 divides 13^(p-1) - 1.

Original entry on oeis.org

2, 863, 1747591

Offset: 1

Alexander Adamchuk, Mar 26 2007

No further terms up to 3.127*10^13.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
C. K. Caldwell, Fermat quotient
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)

Cf. A001220, A014127, A123692, A123693, A128668, A090968, A128669, A039951

Select[Prime[Range[5*10^7]], Mod[ 13^(# - 1) - 1, #^2] == 0 &] (* G. C. Greubel, Jan 18 2018 *)

A045616 Primes p such that 10^(p-1) == 1 (mod p^2).

Original entry on oeis.org

3, 487, 56598313

Offset: 1

Helmut Richter, Dec 11 1999

Primes p such that the decimal fraction 1/p has same period length as 1/p^2, i.e., the multiplicative order of 10 modulo p is the same as the multiplicative order of 10 modulo p^2. [extended by Felix Fröhlich, Feb 05 2017]
No further terms below 1.172*10^14 (as of Feb 2020, cf. Fischer's table).
56598313 was announced in the paper by Brillhart et al. - Helmut Richter, May 17 2004
A265012(A049084(a(n))) = 1. - Reinhard Zumkeller, Nov 30 2015
From Jianing Song, Jun 21 2025: (Start)
The ring of integers of Q(10^(1/k)) is Z[10^(1/k)] if and only if k does not have a prime factor in this sequence. See Theorem 5.3 of the paper of Keith Conrad. For example, we have:
  (1 + 10^(1/3) + 10^(2/3))/3 is an algebraic integer, but it is not in Z[10^(1/3)];
  (1 + 10^(486/487) + 10^(2*486/487) + ... + 10^(486*486/487))/487 is an algebraic integer, but it is not in Z[10^(1/487)];
  (1 + 10^(56598312/56598313) + 10^(2*56598312/56598313) + ... + 10^(56598312*56598312/56598313))/56598313 is an algebraic integer, but it is not in Z[10^(1/56598313)]. (End)

J. Brillhart, J. Tonascia, and P. Weinberger, On the Fermat quotient, pp. 213-222 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
Richard K. Guy, Unsolved Problems in Number Theory, Springer, 2004, A3.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Keith Conrad, The ring of integers in a radical extension.
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2).
Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
Peter L. Montgomery, New solutions of a^(p-1) == 1 (mod p^2), Math. Comp. 61 (1993), 361-363.
Math Overflow, Is the smallest primitive root modulo p a primitive root modulo p^2?, Jun 09 2010.
Helmut Richter, The period length of the decimal expansion of a fraction.
Helmut Richter, The Prime Factors Of 10^486-1.
Siqiong Yao and Akira Toyohara, The length of the repeating decimal, arXiv:2507.01295 [math.NT], 2025. See p. 19.
Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Cf. A001220, A014127, A123692, A212583, A123693, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669, A039951.

Cf. A265012, A049084, A000040.

import Math.NumberTheory.Moduli (powerMod)
a045616 n = a045616_list !! (n-1)
a045616_list = filter
               (\p -> powerMod 10 (p - 1) (p ^ 2) == 1) a000040_list'
-- Reinhard Zumkeller, Nov 30 2015

A045616Q = PrimeQ@# && PowerMod[10, # - 1, #^2] == 1 &; Select[Range[1000000], A045616Q] (* JungHwan Min, Feb 04 2017 *)
Select[Prime[Range[34*10^5]],PowerMod[10,#-1,#^2]==1&] (* Harvey P. Dale, Apr 10 2018 *)

lista(nn) = forprime(p=2, nn, if (Mod(10, p^2)^(p-1)==1, print1(p, ", "))); \\ Michel Marcus, Aug 16 2015

A050217 Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.

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

Offset: 1

Eric W. Weisstein

nonn

Every semiprime in A001567 is in this sequence (see Sierpiński). a(61) = 294409 is the first term having more than two prime factors. See A178997 for super-Poulet numbers having more than two prime factors. - T. D. Noe, Jan 11 2011
Composite numbers n such that 2^d == 2 (mod n) for every d|n. - Thomas Ordowski, Sep 04 2016
Composite numbers n such that 2^p == 2 (mod n) for every prime p|n. - Thomas Ordowski, Sep 06 2016
Composite numbers n = p(1)^e(1)*p(2)^e(2)*...*p(k)^e(k) such that 2^gcd(p(1)-1,p(2)-1,...,p(k)-1) == 1 (mod n). - Thomas Ordowski, Sep 12 2016
Nonsquarefree terms are divisible by the square of a Wieferich prime (see A001220). These include 1194649, 12327121, 5654273717, 26092328809, 129816911251. - Robert Israel, Sep 13 2016
Composite numbers n such that 2^A258409(n) == 1 (mod n). - Thomas Ordowski, Sep 15 2016

W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964, p. 231.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Super-Poulet Numbers
Wikipedia, Super-Poulet number

A214305 is a subsequence.
A065341 is a subsequence. - Thomas Ordowski, Nov 20 2016
Cf. A001220, A001567, A178997.

filter:= = proc(n)
    not isprime(n) and andmap(p -> 2&^p mod n = 2, numtheory:-factorset(n))
end proc:
select(filter, [seq(i,i=3..10^5,2)]); # Robert Israel, Sep 13 2016

Select[Range[1, 110000, 2], !PrimeQ[#] && Union[PowerMod[2, Rest[Divisors[#]], #]] == {2} & ]

is(n)=if(isprime(n), return(0)); fordiv(n,d, if(Mod(2,d)^d!=2, return(0))); n>1 \\ Charles R Greathouse IV, Aug 27 2016

A128668 Primes p such that p^2 divides 17^(p-1) - 1.

Original entry on oeis.org

2, 3, 46021, 48947, 478225523351

Offset: 1

Alexander Adamchuk, Mar 26 2007

Mossinghoff showed that there are no further terms up to 10^14.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)
M. J. Mossinghoff, Wieferich pairs and Barker sequences, Des. Codes Cryptogr. 53 (2009), 149-163.

Cf. A001220, A014127, A123692, A123693, A128667, A090968, A128669, A039951.

Select[Prime[Range[5*10^6]], Mod[ 17^(# - 1) - 1, #^2] == 0 &] (* G. C. Greubel, Jan 18 2018 *)

The prime 478225523351 was found by Richard Fischer on Oct 25 2005

Extension corrected by Jonathan Sondow, Jun 24 2010

A090968 Primes p such that p^2 divides 19^(p-1) - 1.

Original entry on oeis.org

3, 7, 13, 43, 137, 63061489

Offset: 1

Robert G. Wilson v, Feb 27 2004

Primes p such that p divides the Fermat quotient of p (with base 19). The Fermat quotient of p with base a denotes the integer q_p(a) = ( a^(p-1) - 1) / p, where p is a prime which does not divide the integer a. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 20 2005

No further terms up to 3.127*10^13.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 43, p. 17, Ellipses, Paris 2008.
Paulo Ribenboim, The Little Book Of Big Primes, Springer-Verlag, NY 1991, page 170.
Roozbeh Hazrat, Mathematica: A Problem-Centered Approach, Springer 2010, pp. 39, 171. [Harvey P. Dale, Oct 17 2011]

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
C. Caldwell, Fermat quotient
W. Keller and J. Richstein Fermat quotients q_p(a) that are divisible by p

Cf. A001220, A014127, A123692, A123693, A128667, A128668, A128669, A039951, A096082

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ p = NextPrim[p]; If[PowerMod[19, p - 1, p^2] == 1, Print[p]], {n, 1, 2*10^8}]
Select[Prime[Range[4*10^6]],PowerMod[19,#-1,#^2]==1&] (* Harvey P. Dale, Nov 08 2017 *)

A049096 Numbers k such that 2^k + 1 is divisible by a square > 1.

Original entry on oeis.org

3, 9, 10, 15, 21, 27, 30, 33, 39, 45, 50, 51, 55, 57, 63, 68, 69, 70, 75, 78, 81, 87, 90, 93, 99, 105, 110, 111, 117, 123, 129, 130, 135, 141, 147, 150, 153, 159, 165, 170, 171, 177, 182, 183, 189, 190, 195, 201, 204, 207, 210, 213, 219, 225, 230, 231, 234, 237, 243

Offset: 1

Labos Elemer

nonn

Conjecture: lim n -> infinity a(n)/n = C exists and 4 < C < 9/2. There seems to be a sequence of primes p such that p^2 never divides numbers of the form 2^x + 1: the first few are 2, 7, 23, 31. - Benoit Cloitre, Aug 20 2002
That sequence is A072936. - Robert Israel, Nov 20 2015
The first case where 2^n + 1 is divisible by a square that is coprime to n is n = 182 (where 2^182 + 1 is divisible by 1093^2). - Robert Israel, Jul 07 2014
From Robert Israel, Nov 20 2015: (Start)
Numbers n such that gcd(n, 2^n + 1) > 1 or n = k m where k is odd and 2 m is the order of 2 modulo a Wieferich prime.  See link "When p^2 divides 2^n + 1".
If n is in the sequence, then so is k*n for any odd k. (End)
The sequence consists of all odd multiples of { 3, 10, 55, 68, 78, 182, 301, 406, 666, ... }. - M. F. Hasler, Mar 06 2018

			9 is here because 2^9 + 1 = 513 is divisible by 9.
99 is here because 2^99 + 1 = 3^3*19*67*683*5347*20857*242099935645987 is divisible by 9, i.e. is not squarefree.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, When p^2 divides 2^n + 1

Cf. A001220, A049093, A049094, A049095, A072936, A282269, A282270.

Cf. A086982, which is just the same with base b = 10 instead of b = 2.

[n: n in [3..220] | not IsSquarefree(2^n+1)]; // Vincenzo Librandi, Mar 08 2018

remove(n -> numtheory:-issqrfree(2^n+1), [$1..250]); # Robert Israel, Jul 07 2014

Select[Range[243], !SquareFreeQ[2^# + 1] &] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011*)

is(n)=!issquarefree(2^n+1) \\ Altug Alkan, Nov 20 2015

For any a(n+1) - a(n) <= 6 since numbers of form 3^a*(2k+1) a > 0, k >= 0, are in the sequence (2^(3*(2k+1) + 1 is divisible by 9). So are numbers of the form 20k + 10 since 2^(20k+10) + 1 is divisible by 25, 110k + 55 since 2^(110k+55) + 1 is divisible by 11^2, 78 + 156k since 2^(156k+78) + 1 is divisible by 13^2 ... - Benoit Cloitre, Aug 20 2002

More terms from James Sellers, Dec 16 1999
More terms from Vladeta Jovovic, Apr 12 2002
Missing term 182 added by Rainer Rosenthal, Nov 01 2005

cp's OEIS Frontend

A123692 Primes p such that p^2 divides 5^(p-1) - 1.

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A141232 Overpseudoprimes to base 2: composite k such that k = A137576((k-1)/2).

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A123693 Primes p such that p^2 divides 7^(p-1) - 1.

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A077816 Wieferich numbers (1): n > 1 such that 2^A000010(n) == 1 (mod n^2).

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A128667 Primes p such that p^2 divides 13^(p-1) - 1.

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A045616 Primes p such that 10^(p-1) == 1 (mod p^2).

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A050217 Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.

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