A077815 -id:A077815 - cp's OEIS frontend

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

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

A111027 Wieferich primes in base 12.

Original entry on oeis.org

2693, 123653

Offset: 1

Walter Kehowski, Oct 05 2005

I have searched up to the 9 millionth prime, 160481183 and gave up trying to find a third term. The sequence is conjectured to be infinite. If the behavior is similar to base 10, A045616, then the next term could be greater than 2*10^11. In base 12 with X for ten and E for eleven the sequence is [1685, 5E685] so it would be interesting to see if the third term ends in 685 as well. These primes are also the Wieferich numbers in base 12: 12^phi(n) = 1 mod n^2.

Richard Fischer has carried this search to 4.8 * 10^13 (as of January 2014). - John Blythe Dobson, Mar 06 2014

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).

Cf. A001220, A039951, A045616, A077815, A077816, A245529.

WP:=[]: for z from 1 to 1 do for k from 1 to 9000000 do p:=ithprime(k); if 12 &^(p-1) mod p^2 = 1 then WP:=[op(WP),p]; printf("p=%d, ",p); fi; if k mod 10^5 = 0 then printf("k=%d, ",k); fi; od; od; WP;

Select[Prime[Range[1000000]], PowerMod[12, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

12^(p-1) == 1 mod p^2

A319314 Numbers k such that 2^phi(k) == phi(k)^2 (mod k^2).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 10, 12, 384, 640, 768, 896, 960, 24576, 49152, 950272, 1425408, 1572864, 3145728, 10485760, 19398656, 65011712, 100663296, 110057537, 201326592, 220115074, 671088640, 1879048192

Offset: 1

Altug Alkan, Sep 17 2018

Sequence is infinite, i.e., 3*2^(3*(t-1)-(-1)^t) is a term for all t > 0.

Prime terms (5, 110057537, ...) are in A246568 based on case A = +1.

Cf. A077815, A077816, A246568, A292544.

[1] cat [n: n in [1..10^6] | 2^EulerPhi(n) mod n^2 eq EulerPhi(n)^2]; // Vincenzo Librandi, Sep 20 2018

isok(n) = Mod(2, n^2)^eulerphi(n)==eulerphi(n)^2;

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

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A111027 Wieferich primes in base 12.

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A319314 Numbers k such that 2^phi(k) == phi(k)^2 (mod k^2).

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