A282902 -id:A282902 - cp's OEIS frontend

A284044 Largest positive k among all primes p < n such that n^(p-1) == 1 (mod p^k).

1, 1, 2, 1, 2, 2, 3, 2, 1, 1, 2, 1, 1, 1, 4, 3, 3, 1, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 5, 1, 2, 1, 2, 2, 1, 2, 3, 3, 2, 2, 2, 2, 1, 2, 4, 2, 2, 1, 3, 2, 3, 1, 3, 1, 1, 2, 2, 2, 2, 2, 6, 1, 2, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 4, 4, 4, 1, 1, 2, 1, 1, 1, 3

Offset: 3

Felix Fröhlich, Apr 02 2017

nonn

a(n) > 1 iff A255920(n) > 0, i.e., iff n is a term of A273786.

			For n = 7: the maximal exponents k in the congruence 7^(p-1) == 1 (mod p^k) for p = 2, 3, 5 are 1, 1, 2, respectively. Since 2 is the largest exponent among that list, a(7) = 2.

Cf. A255920, A258045, A268310, A273786, A278701, A282902.

a(n) = my(r=1); forprime(p=1, n-1, my(k=1); while(1, if(Mod(n, p^k)^(p-1)!=1, k--; break, k++)); if(k > r, r=k)); r

A305184 Multiplicative order of 2 (mod p^2), where p is the n-th Wieferich prime (A001220).

Original entry on oeis.org

364, 1755

Offset: 1

Felix Fröhlich, May 30 2018

Meissner discovered the congruence 2^364 == 1 (mod 1093^2) and thus proved that 1093 is a Wieferich prime, i.e., a term of A001220 (cf. Meissner, 1913).
Later, Beeger discovered the congruence 2^1755 == 1 (mod 3511^2) and proved that 3511 is also a Wieferich prime (cf. Beeger, 1922).
Let b(n) = (A001220(n)-1)/a(n). Then b(1) = 3 and b(2) = 2.
From the fact that a(1) and a(2) are composite it follows that A001220(1) = 1093 and A001220(2) = 3511 do not divide any terms of A001348 (cf. Dobson).
Curiously, both 364 and 1755 are repdigits in some base. 364 = 444 in base 9 and 1755 = 3333 in base 8. Compare this with Dobson's observation that 1092 and 3510 are 444 in base 16 and 6666 in base 8, respectively (cf. Dobson).

N. G. W. H. Beeger, On a new case of the congruence 2^p-1 == 1 (mod p^2), Messenger of Mathematics 51 (1922), 149-150.
J. B. Dobson, A note on the two known Wieferich primes
W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin, 35 (1913), 663-667. [Annotated scanned copy]

Cf. A001220, A001348, A014664, A243905, A282552, A282902.

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

a(n) = A014664(A000720(A001220(n))) = A243905(A000720(A001220(n))). [Corrected by Jianing Song, Sep 20 2019]

cp's OEIS Frontend

A284044 Largest positive k among all primes p < n such that n^(p-1) == 1 (mod p^k).

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