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

2,1

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)

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Cf. A001220.

Sequence in context: A366688 A054786 A269372 * A033457 A133936 A065350

Adjacent sequences: A282899 A282900 A282901 * A282903 A282904 A282905

Keyword

nonn,tabl

Author

Felix Fröhlich, Feb 24 2017

Status

approved