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A347772 Square array read by antidiagonals downwards: T(n,k) is the smallest prime p not dividing n such that (p-1) / ord_p(n) = k (n>=2, k>=1), or 0 if no such p exists.

%I A347772 #18 Sep 15 2021 10:06:46
%S A347772 3,7,2,43,11,0,113,67,3,2,251,13,0,11,11,31,41,17,13,19,2,1163,61,0,
%T A347772 101,7,3,3,73,883,31,0,5,73,17,2,397,313,0,199,31,29,13,5,7,151,271,
%U A347772 73,827,139,1031,113,0,3,2,331,431,0,569,463,19,251,13,103,7,5,1753,5743,151,487,97,43
%N A347772 Square array read by antidiagonals downwards: T(n,k) is the smallest prime p not dividing n such that (p-1) / ord_p(n) = k (n>=2, k>=1), or 0 if no such p exists.
%F A347772 T(n,k) = 0 if n is square, k is odd, n > 1.
%F A347772 T(n,k) = 0 if n is even square, k is odd.
%F A347772 T(n,k) = 0 if (let n' be the squarefree part (A007913) of n) n' == 1 (mod 4), n' > 1, k is divisible by n', k is odd.
%F A347772 T(27,k) = 0 for k == 4 or 8 (mod 12).
%F A347772 T(n,k) == 1 mod k if nonzero.
%e A347772 Table begins:
%e A347772   n\k |  1   2    3    4     5    6     7    8    9    10    11    12
%e A347772   ----+--------------------------------------------------------------
%e A347772    2  |  3   7   43  113   251   31  1163   73  397   151   331  1753
%e A347772    3  |  2  11   67   13    41   61   883  313  271   431  5743   193
%e A347772    4  |  0   3    0   17     0   31     0   73    0   151     0   433
%e A347772    5  |  2  11   13  101     0  199   827  569  487    31  1453   181
%e A347772    6  | 11  19    7    5    31  139   463   97   37   101   353   241
%e A347772    7  |  2   3   73   29  1031   19    43  113  883   311   353  1453
%e A347772    8  |  3  17   13  113   251    7  1163   89  109   431  1013   577
%e A347772    9  |  2   5    0   13     0   67     0  313    0    41     0    61
%e A347772   10  |  7   3  103   53    11   79   211   41   73   281   353    37
%e A347772   11  |  2   7  193    5   191   19   379  449  199  1301  2531  1549
%e A347772   12  |  5  23   19   37   271   13    29  193  487    11    89   373
%e A347772   ...
%o A347772 (PARI) a(m, n)=forprime(p=2, 2^40, if(gcd(m, p)==1 && znorder(Mod(m, p))==(p-1)/n, return(p)))
%o A347772 is(m, n)=p=core(m); if(p>1 && p%4==1 && n%p==0 && n%2, return(1)); 0
%o A347772 A347772(m, n)=if(is(m, n) || (issquare(m) && n%2 && (m%2==0 || n>1)), 0, a(m, n))
%Y A347772 Row 2: A101208.
%Y A347772 Row 3: A101209.
%Y A347772 Row 10: A054471.
%Y A347772 Column 1: A056619.
%K A347772 nonn,tabl
%O A347772 2,1
%A A347772 _Eric Chen_, Sep 14 2021