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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.

%I A282902 #21 Mar 02 2020 09:39:37
%S A282902 2,6,4,18,20,3,54,100,21,10,162,500,147,110,12,486,2500,1029,1210,156,
%T A282902 8,1458,12500,7203,13310,2028,136,18,4374,62500,50421,146410,26364,
%U A282902 2312,342,11,13122,312500,352947,1610510,342732,39304,6498,253,28,39366,1562500,2470629,17715610,4455516,668168,123462,5819,812,5
%N 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.
%C A282902 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).
%C A282902 From _Robert Israel_, Feb 24 2017: (Start)
%C A282902 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.
%C A282902 a(n,2) = a(n,1) if and only if prime(n) is a Wieferich prime (A001220).
%C A282902 (End)
%H A282902 Robert Israel, <a href="/A282902/b282902.txt">Table of n, a(n) for n = 2..10012</a> (first 142 antidiagonals, flattened)
%e A282902 Array A(n, k) starts
%e A282902    2,   6,   18,     54,     162,      486,      1458
%e A282902    4,  20,  100,    500,    2500,    12500,     62500
%e A282902    3,  21,  147,   1029,    7203,    50421,    352947
%e A282902   10, 110, 1210,  13310,  146410,  1610510,  17715610
%e A282902   12, 156, 2028,  26364,  342732,  4455516,  57921708
%e A282902    8, 136, 2312,  39304,  668168, 11358856, 193100552
%e A282902   18, 342, 6498, 123462, 2345778, 44569782, 846825858
%p A282902 seq(seq(numtheory:-order(2,ithprime(i)^(m-i)),i=2..m-1),m=2..10); # _Robert Israel_, Feb 24 2017
%t A282902 A[n_, k_] := MultiplicativeOrder[2, Prime[n]^k];
%t A282902 Table[A[n-k+1, k], {n, 2, 11}, {k, n-1, 1, -1}] // Flatten (* _Jean-François Alcover_, Mar 02 2020 *)
%o A282902 (PARI) a(n, k) = znorder(Mod(2, prime(n)^k))
%o A282902 array(rows, cols) = for(n=2, rows+1, for(k=1, cols, print1(a(n, k), ", ")); print(""))
%o A282902 array(7, 8) \\ print 7 X 8 array
%Y A282902 Cf. A014664 (column 1), A243905 (column 2).
%Y A282902 Cf. A001220.
%K A282902 nonn,tabl
%O A282902 2,1
%A A282902 _Felix Fröhlich_, Feb 24 2017