A278701 Square array A(n, k) read by antidiagonals downwards: A(n,k) = characteristic function of base-n Wieferich primes: 1 if prime(k) is a base-n Wieferich prime, 0 otherwise, where k runs over the positive integers starting from 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

2

Comments

If A(n, k) = 1, then A(n^t, k) = 1 for every t > 1.

A(n, k) > 0 iff A258045(n, k) > 1.

Column k is the characteristic function of row n of A244249. - Felix Fröhlich, Mar 11 2019

Links

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

Example

A(3, 5) = 1, since prime(5) = 11 is a base-3 Wieferich prime, i.e., 3^10 == 1 (mod 11^2).
A(6, 3) = 0, since prime(3) = 5 is not a base-6 Wieferich prime, i.e., 6^4 =/= 1 (mod 5^2).
Array A(n, k) starts as follows:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0
0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
.
Antidiagonals as a triangular array:
0
0, 0
0, 0, 0
0, 0, 0, 1
0, 0, 0, 0, 0
0, 1, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 1, 1, 1
0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 1, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1

Prog

(PARI) is(n, k) = Mod(n, prime(k)^2)^(prime(k)-1)==1
table(rows, cols) = for(n=2, rows+1, for(k=1, cols, print1(is(n, k), ", ")); print(""))
table(9, 10) \\ print 9 X 10 table
(PARI) first(n) = my(res = vector(n), v = [1, 1]); for(i=1, n, res[i] = is(v[2]+1, v[1]); v=nxt(v)); res
nxt = (v)->if(v[1]==1, [v[2]+1, v[1]], [v[1]-1, v[2]+1]) \\ using is() above. \\ David A. Corneth, Mar 11 2019

Crossrefs

Cf. A244249, A258045.

Sequence in context: A044940 A284261 A037825 * A188300 A353475 A277154

Adjacent sequences: A278698 A278699 A278700 * A278702 A278703 A278704

Keyword

nonn,tabl

Author

Felix Fröhlich, Dec 05 2016

Extensions

More terms from Felix Fröhlich, Mar 11 2019

Status

approved