A123692 -id:A123692 - cp's OEIS frontend

A243010 Pseudoprimes to base 5 that are not squarefree.

4, 124, 11476, 59356, 80476, 91636, 250876, 261964, 482516, 1385836, 1926676, 2428084, 2589796, 3743476, 4101796, 6797764, 9155476, 10701076, 10743436, 11263396, 13799836, 13859956, 15570556, 20396476

Offset: 1

Felix Fröhlich, Aug 18 2014

nonn

Any term is divisible by the square of a base 5 Wieferich prime (A123692).

Intersection of A005936 and A013929. - Michel Marcus, Aug 21 2014

Amiram Eldar, Table of n, a(n) for n = 1..879 (terms below 10^12)

Cf. A005936, A123692, A158358, A243089, A243090, A244065.

forcomposite(n=1, 1e9, if(Mod(5, 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

A342391 Odd numbers k such that the ring of integers of Q(5^(1/k)) is not Z[5^(1/k)].

Original entry on oeis.org

20771, 40487, 62313, 103855, 121461, 145397, 186939, 202435, 228481, 270023, 283409, 311565, 353107, 364383, 394649, 436191, 445357, 477733, 519275, 526331, 560817, 602359, 607305, 643901, 685443, 688279, 726985, 768527, 769253, 810069, 850227, 851611, 893153, 931201, 934695, 976237

Offset: 1

Jianing Song, Mar 10 2021

nonn

For k > 1, a != 1 being a squarefree number (a != -1 unless k is a power of 2), then the ring of integers of Q(a^(1/k)) is Z[a^(1/k)] if and only if: for every p dividing k, we have a^(p-1) !== 1 (mod p^2). In other words, O_Q(a^(1/k)) = Z[a^(1/k)] if and only if none of the prime factors of k is a Wieferich prime of base a. See Theorem 5.3 of the paper of Keith Conrad.
In general, if a^d == 1 (mod p^2) for some d|(p-1), then it is easy to show that x = (1 + a^(d/p) + a^(2*d/p) + ... + a^((p-1)*d/p))/p is an algebraic integer not in Z[a^(1/p)].
Here a = 5. Since 2 is Wieferich prime of base 5, for all even k we have O_Q(a^(1/k)) != Z[a^(1/k)]. There are only 6 other known Wieferich primes of base 5 (A123692) up to 9.7*10^14.

			5^10385 == 1 (mod 20771^2), so x = (1 + 5^(10385/20771) + 5^(2*10385/20771) + ... + 5^(20770*10385/20771))/20771 is an algebraic integer not in Z[5^(1/20771)].

Jianing Song, Table of n, a(n) for n = 1..7286 (all terms up to 2*10^8).
Keith Conrad, The ring of integers in a radical extension

Cf. A123692, A342390.

Wieferich_up_to_n(lim) = my(v=[]); forprime(p=2, lim, if(Mod(5,p^2)^(p-1)==1, v=concat(v,p))); v
Up_to_n(lim) = my(pv=Wieferich_up_to_n(lim), list=[]); for(i=2, #pv, my(p=pv[i]); forstep(k=1, lim\p, 2, list=concat(list, k*p))); list=Set(list); list \\ corrected by Jianing Song, Mar 21 2022

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", ")))

A331426 Prime numbers p such that p^2 divides 37^(p-1) - 1.

Original entry on oeis.org

2, 3, 77867, 76407520781

Offset: 1

Seiichi Manyama, Jan 16 2020

a(4) from Fischer link. - M. F. Hasler, 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), A331424 (b=31), this sequence (b=37), A331427 (b=41).

Cf. A039951.

Select[Range[10^5], PrimeQ[#] && PowerMod[37, # - 1, #^2] == 1 &] (* Amiram Eldar, May 05 2021 *)

forprime(p=2, 1e8, if(Mod(37, p^2)^(p-1)==1, print1(p", ")))

A331427 Prime numbers p such that p^2 divides 41^(p-1) - 1.

Original entry on oeis.org

2, 29, 1025273, 138200401

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), A331424 (b=31), A331426 (b=37), this sequence (b=41).

Cf. A039951.

Select[Range[1.1*10^6], PrimeQ[#] && PowerMod[41, # - 1, #^2] == 1 &] (* Amiram Eldar, May 05 2021 *)

forprime(p=2, 1e8, if(Mod(41, p^2)^(p-1)==1, print1(p", ")))

A250206 Least base b > 1 such that b^A000010(n) = 1 (mod n^2).

Original entry on oeis.org

2, 5, 8, 7, 7, 17, 18, 15, 26, 7, 3, 17, 19, 19, 26, 31, 38, 53, 28, 7, 19, 3, 28, 17, 57, 19, 80, 19, 14, 107, 115, 63, 118, 65, 18, 53, 18, 69, 19, 7, 51, 19, 19, 3, 26, 63, 53, 17, 18, 57, 134, 19, 338, 161, 3, 31, 28, 41, 53, 107, 264, 115, 19, 127, 99, 161, 143, 65, 28, 99, 11, 55

Offset: 1

Eric Chen, Feb 21 2015

nonn

a(n) = least base b > 1 such that n is a Wieferich number (see A077816).
At least, b = n^2+1 can satisfy this equation, so a(n) is defined for all n.
Least Wieferich number (>1) to base n: 2, 1093, 11, 1093, 2, 66161, 4, 3, 2, 3, 71, 2693, 2, 29, 4, 1093, 2, 5, 3, 281, 2, 13, 4, 5, 2, ...; each is a prime or 4. It is 4 if and only if n mod 72 is in the set {7, 15, 23, 31, 39, 47, 63}.
Does every natural number (>1) appear in this sequence? If yes, do they appear infinitely many times?
For prime n, a(n) = A185103(n), does there exist any composite n such that a(n) = A185103(n)?

			a(30) = 107 since A000010(30) = 8, 30^2 = 900, and 107 is the least base b > 1 such that b^8 = 1 (mod 900).

Eric Chen, Table of n, a(n) for n = 1..1000

Cf. A039678, A185103, A125636, A039951, A247154, A001220, A014127, A123692, A212583, A123693, A045616, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669, A077816, A242958, A242959, A241978, A242960, A241977, A253016, A245529.

f[n_] := Block[{b = 2, m = EulerPhi[n]}, While[ PowerMod[b, m, n^2] != 1, b++]; b]; f[1] = 2; Array[f, 72] (* Robert G. Wilson v, Feb 28 2015 *)

a(n)=for(k=2,2^24,if((k^eulerphi(n))%(n^2)==1, return(k)))

a(prime(n)) = A039678(n) = A185103(prime(n)).
a(A077816(n)) = 2.
a(A242958(n)) <= 3.

cp's OEIS Frontend

A243010 Pseudoprimes to base 5 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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A342391 Odd numbers k such that the ring of integers of Q(5^(1/k)) is not Z[5^(1/k)].

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

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A331426 Prime numbers p such that p^2 divides 37^(p-1) - 1.

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