A258045 Table T(b, m) of largest exponents k such that for p = prime(m) and base b > 1 the congruence b^(p-1) == 1 (mod p^k) is satisfied, or 0 if no such k exists, read by antidiagonals (downwards).

0, 1, 1, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

2,10

Comments

a(n) > 1 if b appears in row k, column n of the table in A257833 for k > 1 and n > 1.

Links

Table of n, a(n) for n=2..88.

Formula

a(n, m) = T(m+1, n-m), n >=2, m = 1, 2, ..., n-1. - Wolfdieter Lang, Jun 29 2015

Example

T(3, 5) = 2, because the largest Wieferich exponent of prime(5) = 11 in base 3 is 2.
Table starts
b=2:  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=3:  1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=4:  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=5:  2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=6:  0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=7:  1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=8:  0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=9:  3, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=10: 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=11: 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=12: 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=13: 2, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=14: 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1 ...
b=15: 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=16: 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=17: 4, 2, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1 ...
b=18: 0, 0, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1 ...
b=19: 1, 2, 1, 3, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2 ...
b=20: 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
....
The triangle a(n ,m) begins:
  m 1 2 3 4 5 6 7 8 9 10 11 ...
n
2   0
3   1 1
4   1 0 0
5   1 1 1 2
6   1 1 1 1 0
7   1 2 1 0 0 1
8   1 1 1 1 1 1 0
9   1 1 1 1 1 2 2 3
10  1 1 1 1 1 0 1 0 0
11  1 1 1 1 1 1 1 1 2  1
12  1 1 1 1 1 1 1 1 0  1  0
...

Prog

(PARI) for(b=2, 20, forprime(p=1, 70, k=0; while(Mod(b, p^k)^(p-1)==1, k++); if(k > 0, k--); print1(k, ", ")); print(""))

Crossrefs

Cf. A001220, A257833.

Sequence in context: A037818 A087116 A033264 * A239302 A256983 A330720

Adjacent sequences: A258042 A258043 A258044 * A258046 A258047 A258048

Keyword

nonn,tabl

Author

Felix Fröhlich, May 26 2015

Extensions

Edited by Wolfdieter Lang, Jun 29 2015

Status

approved