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
Examples
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 ...
Programs
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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(""))
Formula
a(n, m) = T(m+1, n-m), n >=2, m = 1, 2, ..., n-1. - Wolfdieter Lang, Jun 29 2015
Extensions
Edited by Wolfdieter Lang, Jun 29 2015
Comments