A282902 Square array A(n, k) read by antidiagonals downwards: multiplicative order of 2 modulo prime(n)^k, where k runs over the positive integers.

2, 6, 4, 18, 20, 3, 54, 100, 21, 10, 162, 500, 147, 110, 12, 486, 2500, 1029, 1210, 156, 8, 1458, 12500, 7203, 13310, 2028, 136, 18, 4374, 62500, 50421, 146410, 26364, 2312, 342, 11, 13122, 312500, 352947, 1610510, 342732, 39304, 6498, 253, 28, 39366, 1562500, 2470629, 17715610, 4455516, 668168, 123462, 5819, 812, 5

Offset: 2

Author

Felix Fröhlich, Feb 24 2017

Keywords

• nonn
• tabl

Comments

The number of initial terms in row n with constant values is equal to the highest value of x such that p = prime(n) satisfies 2^(p-1) == 1 (mod p^x).

From Robert Israel, Feb 24 2017: (Start)

a(n,k+1) is either a(n,k) or a(n,k)*prime(n). If it is a(n,k)*prime(n), then a(n,k+j) = a(n,k)*prime(n)^j for all j>=1.

a(n,2) = a(n,1) if and only if prime(n) is a Wieferich prime (A001220).

(End)

Examples

			Array A(n, k) starts
   2,   6,   18,     54,     162,      486,      1458
   4,  20,  100,    500,    2500,    12500,     62500
   3,  21,  147,   1029,    7203,    50421,    352947
  10, 110, 1210,  13310,  146410,  1610510,  17715610
  12, 156, 2028,  26364,  342732,  4455516,  57921708
   8, 136, 2312,  39304,  668168, 11358856, 193100552
  18, 342, 6498, 123462, 2345778, 44569782, 846825858
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10012 (first 142 antidiagonals, flattened)

Crossrefs

Cf. A014664 (column 1), A243905 (column 2).

Cf. A001220.

Programs

Maple

seq(seq(numtheory:-order(2,ithprime(i)^(m-i)),i=2..m-1),m=2..10); # Robert Israel, Feb 24 2017

Mathematica

A[n_, k_] := MultiplicativeOrder[2, Prime[n]^k];
Table[A[n-k+1, k], {n, 2, 11}, {k, n-1, 1, -1}] // Flatten (* Jean-François Alcover, Mar 02 2020 *)

PARI

a(n, k) = znorder(Mod(2, prime(n)^k))
array(rows, cols) = for(n=2, rows+1, for(k=1, cols, print1(a(n, k), ", ")); print(""))
array(7, 8) \\ print 7 X 8 array