A123693 -id:A123693 - cp's OEIS frontend

A242960 Numbers n such that 7^A000010(n) == 1 (mod n^2).

4, 5, 10, 20, 40, 80, 491531, 983062, 1966124, 2457655, 3932248, 4915310, 6389903, 9339089, 9830620, 12288275, 12779806, 18678178, 19169709, 19661240, 24576550, 25559612, 28017267, 31949515, 37356356, 38339418, 39322480, 46695445, 49153100, 51119224, 56034534

Offset: 1

Felix Fröhlich, May 27 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..425

Cf. A077816, A242959, A000010. All primes in this sequence are in A123693.

for(n=2, 10^9, if(Mod(7, n^2)^(eulerphi(n))==1, print1(n, ", ")))

Terms a(23) and beyond from Giovanni Resta, Jan 24 2020

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, ", ")));

A243089 Pseudoprimes to base 7 that are not squarefree.

Original entry on oeis.org

25, 325, 1825, 4525, 4825, 10225, 12025, 16725, 20425, 30025, 35425, 58825, 177025, 216525, 265525, 352225, 526825, 611425, 675925, 710425, 717025, 746425, 772525, 784225, 834025, 877825, 1125825, 1126225, 1439425, 1491025, 1579225, 1935025, 1973425, 2176525

Offset: 1

Felix Fröhlich, Aug 18 2014

nonn

Any term is divisible by the square of a base 7 Wieferich prime (A123693).

Intersection of A005938 and A013929. - Michel Marcus, Aug 21 2014

Amiram Eldar, Table of n, a(n) for n = 1..3095 (terms below 10^12)

Cf. A005938, A123693, A158358, A243010, A243090, A244065.

forcomposite(n=1, 1e9, if(Mod(7, n)^(n-1)==1, if(!issquarefree(n), print1(n, ", "))))

A298951 Wieferich primes to base 22.

Original entry on oeis.org

13, 673, 1595813, 492366587, 9809862296159

Offset: 1

Tim Johannes Ohrtmann, Jan 30 2018

Prime numbers p such that p^2 divides 22^(p-1) - 1.
Next term, if it exists, is larger than 8.72*10^13.
492366587 was found by Montgomery (cf. Montgomery, 1993). - Felix Fröhlich, Jan 30 2018

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Richard Fischer, Fermatquotient B^(P-1) == 1 (mod P^2)
P. L. Montgomery, New Solutions of a^p-1 == 1 (mod p^2), Mathematics of Computation, Vol. 61, No. 203 (1993), 361-363.
Wikipedia, Wieferich prime

Wieferich primes to base b: A001220 (b=2), A014127 (b=3), A123692 (b=5), A212583 (b=6), A123693 (b=7), A045616 (b=10), A111027 (b=12), A128667 (b=13), A234810 (b=14), A242741 (b=15), A128668 (b=17), A244260 (b=18), A090968 (b=19), A242982 (b=20), this sequence (b=22), A128669 (b=23), A306255 (b=26), A306256 (b=30).

forprime(p=1, , if(Mod(22, p^2)^(p-1)==1, print1(p, ", ")))

A306255 Wieferich primes to base 26.

Original entry on oeis.org

3, 5, 71, 486999673, 6695256707

Offset: 1

Jianing Song, Feb 01 2019

Prime numbers p such that p^2 divides 26^(p-1) - 1.

No more terms up to 9.8*10^13.

Richard Fischer, Fermatquotient B^(P-1) == 1 (mod P^2)
P. L. Montgomery, New Solutions of a^p-1 == 1 (mod p^2), Mathematics of Computation, Vol. 61, No. 203 (1993), 361-363.
Wikipedia, Wieferich prime

Wieferich primes to base b: A001220 (b=2), A014127 (b=3), A123692 (b=5), A212583 (b=6), A123693 (b=7), A045616 (b=10), A111027 (b=12), A128667 (b=13), A234810 (b=14), A242741 (b=15), A128668 (b=17), A244260 (b=18), A090968 (b=19), A242982 (b=20), A298951 (b=22), A128669 (b=23), this sequence (b=26), A306256 (b=30).

Select[Prime[Range[26*10^6]],PowerMod[26,#-1,#^2]==1&] (* The program generates the first 4 terms of the sequence. *) (* Harvey P. Dale, Aug 23 2024 *)

forprime(p=2, , if(Mod(26, p^2)^(p-1)==1, print1(p, ", ")))

A306256 Wieferich primes to base 30.

Original entry on oeis.org

7, 160541, 94727075783

Offset: 1

Jianing Song, Feb 01 2019

Prime numbers p such that p^2 divides 30^(p-1) - 1.

No more terms up to 9.8*10^13.

Richard Fischer, Fermatquotient B^(P-1) == 1 (mod P^2)
P. L. Montgomery, New Solutions of a^p-1 == 1 (mod p^2), Mathematics of Computation, Vol. 61, No. 203 (1993), 361-363.
Wikipedia, Wieferich prime

Wieferich primes to base b: A001220 (b=2), A014127 (b=3), A123692 (b=5), A212583 (b=6), A123693 (b=7), A045616 (b=10), A111027 (b=12), A128667 (b=13), A234810 (b=14), A242741 (b=15), A128668 (b=17), A244260 (b=18), A090968 (b=19), A242982 (b=20), A298951 (b=22), A128669 (b=23), A306255 (b=26), this sequence (b=30).

forprime(p=2, , if(Mod(30, p^2)^(p-1)==1, print1(p, ", ")))

A247072 Smallest Wieferich prime (> sqrt(n)) in base n.

Original entry on oeis.org

2, 1093, 11, 1093, 20771, 66161, 5, 3, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281

Offset: 1

Eric Chen, Nov 16 2014

a(n) = Smallest prime such that n appears in A143548. - Eric Chen, Nov 26 2014
The square of a(n) is the smallest squared prime that is a pseudoprime (> n) in base n; for example, a(2) = 1093, and 1093^2 = 1194649 is the smallest squared prime that is pseudoprime in base 2. - Eric Chen, Nov 26 2014
Is a(n) defined for all n? - Eric Chen, Nov 26 2014
Does every prime appear in this sequence? - Eric Chen, Nov 26 2014
a(22)..a(28) = {13, 13, 5, 20771, 71, 11, 19}, a(30)..a(46) = {7, 7, 1093, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 103, 229, 1283, 829}, a(48)..a(49) = {7, 491531}, a(51)..a(60) = {41, 461, 47, 19, 30109, 647, 47699, 131, 2777, 29}, a(62)..a(71) = {19, 23, 1093, 17, 89351671, 47, 19, 19, 13, 47}, a(74)..a(81) = {1251922253819, 17, 37, 32687, 43, 263, 13, 11}, a(83)..a(100) = {4871, 163, 11779, 68239, 1999, 2535619637, 13, 6590291053, 293, 727, 509, 11, 2137, 109, 2914393, 28627, 13, 487}; a(n) is currently unknown for n = {21, 29, 47, 50, 61, 72, 73, 82, 126, 132, 154, 186, 187, 188, 200, 203, 222, 231, 237, 301, 304, 309, 311, 327, 335, 347, 351, 355, 357, 435, 441, 454, 458, 496, 541, 542, 546, 554, 570, 593, 609, 610, 639, 640, 654, 662, 668, 674, 692, 697, 698, 701, 718, 724, 725, 727, 733, 743, 760, 772, 775, 777, 784, 798, 807, 808, 812, 829, 841, 858, 860, 871, 883, 912, 919, 944, 980, 983, 986, ...}. - Eric Chen, Nov 26 2014
a(21) > 3.4 * 10^13. - Eric Chen, Nov 26 2014

			a(12) = 2693 because the Wieferich primes to base 12 are 2693, 123653, ..., and 2693 is greater than sqrt(12), so a(12) = 2693.
a(17) = 46021 because the Wieferich primes to base 17 are 2, 3, 46021, 48947, 478225523351, ..., but neither 2 nor 3 is greater than sqrt(17), so a(17) = 46021.

Eric Chen, Table of known a(n) up to a(1000)
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)
Wikipedia, Wieferich prime

Cf. A039951, A096082, A174422, A178871.

Cf. A001220, A014127, A123692, A212583, A123693, A045616, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669.

a247072[n_] := Block[{p = Int[Sqrt[n]]+1}, While[!PrimeQ[p] || [p < 10^8 && PowerMod[n, p - 1, p^2] != 1], p++]; If[p == 10^8, 0, p]]; Table[ a247072[n], {n, 100}] (* Eric Chen, Nov 27 2014 *)

a(n)=forprime(p=sqrtint(n)+1,,if(Mod(n^(p-1),p^2)==1,return(p)))
n=1; while(n<101, print1(a(n), ", "); n++) \\ Charles R Greathouse IV, Nov 16 2014

A331424 Prime numbers p such that p^2 divides 31^(p-1) - 1.

Original entry on oeis.org

7, 79, 6451, 2806861

Offset: 1

Seiichi Manyama, Jan 16 2020

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

Richard Fischer, Fermatquotienten von 2 bis 1052, Dec 19 2019.
Wikipedia, Wieferich prime

Wieferich primes to base b: A001220 (b=2), A014127 (b=3), A123692 (b=5), A123693 (b=7), A128667 (b=13), A128668 (b=17), A090968 (b=19), A128669 (b=23), this sequence (b=31), A331426 (b=37), A331427 (b=41).

Cf. A039951.

Select[Range[3*10^6], PrimeQ[#] && PowerMod[31, # - 1, #^2] == 1 &] (* Amiram Eldar, May 05 2021 *)

forprime(p=2, 1e8, if(Mod(31, p^2)^(p-1)==1, print1(p", ")))

cp's OEIS Frontend

A242960 Numbers n such that 7^A000010(n) == 1 (mod n^2).

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

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

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

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A243089 Pseudoprimes to base 7 that are not squarefree.

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A298951 Wieferich primes to base 22.

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A306255 Wieferich primes to base 26.

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A306256 Wieferich primes to base 30.

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A247072 Smallest Wieferich prime (> sqrt(n)) in base n.

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