A123693 -id:A123693 - cp's OEIS frontend

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

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.

A259909 n-th Wieferich prime to base prime(n), i.e., primes p such that p is the n-th solution of the congruence (prime(n))^(p-1) == 1 (mod p^2).

Original entry on oeis.org

1093, 1006003, 40487

Offset: 1

Felix Fröhlich, Jul 07 2015

Main diagonal of table T(b, p) of Wieferich primes p to prime bases b (that table is not yet in the OEIS as a sequence).
a(4), if it exists, corresponds to A123693(4) and is larger than 9.7*10^14 (cf. Dorais, Klyve, 2011).
a(5), if it exists, corresponds to the 5th base-11 Wieferich prime and is larger than approximately 5.9*10^13 (cf. Fischer).
a(6), if it exists, corresponds to A128667(6) and is larger than approximately 5.9*10^13 (cf. Fischer).

			a(1) = A001220(1) = 1093.
a(2) = A014127(2) = 1006003.
a(3) = A123692(3) = 40487.

W. Keller, Prime solutions p of a^p-1 = 1 (mod p2) for prime bases a, Abstracts Amer. Math. Soc., 19 (1998), 394.

M. Aaltonen and K. Inkeri, Catalan's equation x^p - y^q and related congruences, Mathematics of Computation, Vol. 56 No. 193 (1991), 359-370.
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.
R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)
W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p (Cached copy at the Wayback Machine).
K. E. Kloss, Some Number-Theoretic Calculations, J. Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (1965), 335-336.

Cf. A001220, A014127, A123692, A123693, A174422, A178871.

a(n) = my(i=0, p=2); while(i < n, if(Mod(prime(n), p^2)^(p-1)==1, i++; if(i==n, break({1}))); p=nextprime(p+1)); p

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

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

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

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A259909 n-th Wieferich prime to base prime(n), i.e., primes p such that p is the n-th solution of the congruence (prime(n))^(p-1) == 1 (mod p^2).

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