A014127 -id:A014127 - cp's OEIS frontend

A238490 Odd primes p that divide a Lucas quotient studied by H. C. Williams: A001353(p - (3/p))/p, where (3/p) is a Jacobi symbol.

103, 2297860813

Offset: 1

John Blythe Dobson, Mar 28 2014

The condition for an odd prime p to be a member of this sequence is that p^2 divides A001353(p - (3/p)).
Neither this quotient, nor the Lucas sequence U(4, 1) on which it is based, has a common name; but its fundamental discriminant of 3 places it between the quotient based on the Pell sequence U(2, -1) with discriminant 2 (A000129), and that based on the Fibonacci sequence U(1, -1) with discriminant 5 (A000045). Values of p dividing the Pell quotient will be found under A238736, while for the Fibonacci quotient it is known that there is no such p < 9.7*10^14.
The interest in this family of number-theoretic quotients derives from H. C. Williams, "Some formulas concerning the fundamental unit of a real quadratic field," p. 440, which proves a formula connecting the present quotient with the Fermat quotient base 2 (A007663), the Fermat quotient base 3 (A146211), and the harmonic number H(floor(p/12)) (see the Formula section below). As is well known, the vanishing of each of these Fermat quotients is a necessary condition for the failure of the first case of Fermat's Last Theorem (see discussions under A001220 and A014127); and a corresponding result concerning this type of harmonic number was proved by Dilcher and Skula. Thus, the vanishing mod p of the quotient based on U(4, 1) is also a necessary condition for the failure of the first case of Fermat's Last Theorem.
The pioneering computation for this quotient appears to be that of Elsenhans and Jahnel, "The Fibonacci sequence modulo p^2," p. 5, who report 103 as the only value of a(n) < 10^9. Extending the search to p < 2.5*10^10 has found only one further solution, 2297860813.
Let LucasQuotient(p) = A001353(p - (3/p))/p, q_2 = (2^(p-1) - 1)/p = A007663(p) be the corresponding Fermat quotient of base 2, q_3 = (3^(p-1) - 1)/p = A146211(p) be the corresponding Fermat quotient of base 3, H(floor(p/12)) be a harmonic number. Then Williams (1991) shows that 6*(3/p)*LucasQuotient(p) == -6*q_2 - 3*q_3 - 2*H(floor(p/12)) (mod p).
Also with an initial 2, primes p such that p^2 divides A001353(p - Kronecker(12,p)) (note that 12 is the discriminant of the characteristic polynomial of A001353, x^2 - 4x + 1). - Jianing Song, Jul 28 2018

			LucasQuotient(103) = 103*851367555454046677501642274766916900879231854719584128208.

John Blythe Dobson, Table of n, a(n) for n = 1..2
Karl Dilcher and Ladislav Skula, A new criterion for the first case of Fermat's Last Theorem, Mathematics of Computation, 64 (1995) 363-392.
Andreas-Stephan Elsenhans and Jörg Jahnel, The Fibonacci sequence modulo p^2 -- An investigation by computer for p < 1014, arxiv 1006.0824 [math.NT], 2010.
H. C. Williams, Some formulas concerning the fundamental unit of a real quadratic field, Discrete Mathematics, 92 (1991), 431-440.

Cf. A001353, A238736.

The following criteria are equivalent:
PrimeQ[p] &&
  Mod[(MatrixPower[{{1,2},{1,3}}, p-JacobiSymbol[3,p]-1].{{1},{1}})[[2,1]], p^2]==0
PrimeQ[p] && Mod[Last[LinearRecurrence[{4,-1},{0,1}, p-JacobiSymbol[3,p]+1]], p^2]==0

isprime(p) && (Mod([2, 2; 1, 0], p^2)^(p-kronecker(3, p)))[2, 1]==0 \\ This test, which was used to find the second member of this sequence, is based on the test for A238736 devised by Charles R Greathouse IV

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, ", ")))

A178815 First base of a nonzero Fermat quotient mod the n-th prime.

Original entry on oeis.org

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Jonathan Sondow, Jun 17 2010, Jun 24 2010, Jun 25 2010

nonn

First number m coprime to p = p_n such that p does not divide q_p(m), where q_p(m) = (m^(p-1) - 1)/p is the Fermat quotient of p to the base m.
It is known that a(n) = O((log p_n)^2) as n -> oo. It is conjectured that a(n) = 3 if p_n is a Wieferich prime. See Section 1.1 in Ostafe-Shparlinski (2010).
Additional comments, references, links, and cross-refs are in A001220.
a(n) > 3 iff prime(n) is a term of both A001220 and A014127, i.e., iff A240987(n) = 2. - Felix Fröhlich, Jul 09 2016

			p_1 = 2 and 2^2 divides 1^(2-1) - 1 = 0 but not 3^(2-1) - 1 = 2, so a(1) = 3.
p_4 = 7 and 7^2 does not divide 2^(7-1) - 1 = 63, so a(4) = 2.
p_183 = 1093 and 1093^2 divides 2^1092 - 1 but not 3^1092 - 1, so a(183) = 3.
Similarly, p_490 = 3511 and a(490) = 3. See A001220.

A. Ostafe and I. Shparlinski, Pseudorandomness and Dynamics of Fermat Quotients, arXiv:1001.1504 [math.NT], 2010.
Wikipedia, Generalized Wieferich primes

Cf. A001220, A178844.

Table[b = 2; While[PowerMod[b, Prime[n] - 1, #^2] == 1 || GCD[b, #] > 1, b++] &@ Prime@ n; b, {n, 120}] (* Michael De Vlieger, Jul 09 2016 *)

a(n) = my(b=2, p=prime(n)); while(Mod(b, p^2)^(p-1)==1 || gcd(b, p) > 1, b++); b \\ Felix Fröhlich, Jul 09 2016

a(n) = 2 if n > 1 and p_n is not a Wieferich prime A001220.
a(n) > 2 if p_n is a Wieferich prime.
A178844(n) = ((a(n)^(p-1) - 1)/p) mod p, where p = p_n.

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

A260377 Primes p such that A001221(p-1)^(p-1) == 1 (mod p^2).

Original entry on oeis.org

3, 5, 17, 257, 1093, 3511, 65537, 1006003

Offset: 1

Felix Fröhlich, Jul 23 2015

No further terms up to 10^9.

All currently (as of 2015-07-23) known terms of A001220 and A019434 are in the sequence, as is A014127(2), but A014127(1) is not.

			A001221(1092) == 4 and 4^1092 == 1 (mod 1093^2), so 1093 is a term of the sequence.

Cf. A001220, A014127, A019434, A244546.

forprime(p=1, 1e9, if(Mod(omega(p-1), p^2)^(p-1)==1, print1(p, ", ")))

A179078 a(n) = ((3^p - 3)/p) mod p where p is n-th prime.

Original entry on oeis.org

1, 2, 3, 4, 0, 11, 13, 16, 5, 16, 20, 17, 22, 6, 33, 16, 5, 39, 45, 25, 5, 4, 26, 72, 21, 53, 43, 80, 85, 12, 53, 94, 54, 135, 113, 132, 125, 32, 34, 163, 100, 147, 52, 61, 84, 46, 54, 80, 122, 103, 83, 43, 109, 87, 127, 125, 239, 129, 63, 98, 160, 223, 29, 82, 3, 68, 288, 322

Offset: 1

Artur Jasinski, Jun 28 2010

nonn

a(n) = 0 where n=5 (p=11) and n=78940 (p=1006003) see A014127.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014127, A179077, A377669.

f:= p -> (3&^p-3 mod p^2)/p:
seq(f(ithprime(i)),i=1..100); # Robert Israel, Nov 03 2024

aa = {}; Do[AppendTo[aa, Mod[(3^Prime[n] - 3)/Prime[n], Prime[n]]], {n, 1, 100}]; aa (* Artur Jasinski *)

A240987 (2^(p-1) modulo p^2) + (3^(p-1) modulo p^2), where p = prime(n).

Original entry on oeis.org

5, 4, 22, 58, 57, 145, 393, 401, 784, 466, 715, 705, 1806, 1163, 2587, 3129, 2893, 2991, 1677, 2416, 5988, 5769, 9298, 2672, 6210, 17879, 14628, 11879, 18314, 9833, 9908, 12054, 9729, 10427, 34719, 15102, 27634

Offset: 1

Felix Fröhlich, Aug 06 2014

nonn

A value of 2 would indicate a prime that is Wieferich to bases 2 and 3 (i.e., a term of both A001220 and A014127). No such prime is currently known.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
J. B. Dobson, On Lerch's formula for the Fermat quotient, arXiv:1103.3907 [math.NT], 2011-2014.

Cf. A001220, A014127.

map(p -> (2 &^ (p-1) mod p^2) + (3 &^ (p-1) mod p^2), select(isprime,[2,seq(2*i+1,i=1..1000)])); # Robert Israel, Aug 11 2014

Table[p = Prime[n]; PowerMod[2, p-1, p^2] + PowerMod[3, p-1, p^2], {n, 40}] (* Jean-François Alcover, Sep 19 2018 *)

forprime(p=2, 1e2, a=2^(p-1)%p^2; b=3^(p-1)%p^2; print1(a+b, ", "))

A306448 Pseudoprimes to base 9 that are not squarefree.

Original entry on oeis.org

4, 8, 28, 52, 121, 364, 532, 616, 1036, 1288, 3052, 3751, 4376, 4636, 4961, 5356, 6364, 7381, 8744, 11011, 11476, 12124, 15964, 19096, 19684, 21196, 21736, 24388, 26596, 29161, 31876, 32791, 37576, 40132, 45676, 47972, 53092, 61831, 67276, 72136, 80476, 80956, 86296

Offset: 1

Jianing Song, Feb 16 2019

nonn

Numbers k that are not squarefree and satisfy 9^(k-1) == 1 (mod k).
Any term is divisible by the square of a base-9 Wieferich prime ({2} U {base-3 Wieferich primes} = {2} U A014127 = {2, 11, 1006003, ...}).
Intersection of A020138 and A013929.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Pseudoprimes to base b that are not squarefree: A158358 (b=2), A244065 (b=3), A243010 (b=5), A243089 (b=7), A243090 (b=8), this sequence (b=9), A306449 (b=10).

Cf. also A014127, A020138, A013929.

for(n=1, 10^5, if(Mod(9, n)^(n-1)==1 && !issquarefree(n), print1(n, ", ")))

cp's OEIS Frontend

A238490 Odd primes p that divide a Lucas quotient studied by H. C. Williams: A001353(p - (3/p))/p, where (3/p) is a Jacobi symbol.

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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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A178815 First base of a nonzero Fermat quotient mod the n-th prime.

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Formula

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

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A260377 Primes p such that A001221(p-1)^(p-1) == 1 (mod p^2).

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A179078 a(n) = ((3^p - 3)/p) mod p where p is n-th prime.

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