A277915 A(n,k) is the n-th number m such that a nontrivial prime(k)-th root of unity modulo m exists; square array A(n,k), n>=1, k>=1, read by antidiagonals.

8, 7, 12, 11, 9, 15, 29, 22, 13, 16, 23, 43, 25, 14, 20, 53, 46, 49, 31, 18, 21, 103, 79, 67, 58, 33, 19, 24, 191, 137, 106, 69, 71, 41, 21, 28, 47, 229, 206, 131, 89, 86, 44, 26, 30, 59, 94, 361, 239, 157, 92, 87, 50, 27, 32, 311, 118, 139, 382, 274, 158, 115, 98, 55, 28, 33

Offset: 1

Author

Alois P. Heinz, Nov 03 2016

Keywords

• nonn
• look
• tabl

Comments

The trivial square roots of unity modulo m are {1, m-1} and for an odd prime p the trivial p-th root of unity modulo m is 1.

There is no prime in the first column.

Column k>1 contains prime(k)^2.

Examples

			Square array A(n,k) begins:
:  8,  7, 11, 29,  23,  53, 103, 191, ...
: 12,  9, 22, 43,  46,  79, 137, 229, ...
: 15, 13, 25, 49,  67, 106, 206, 361, ...
: 16, 14, 31, 58,  69, 131, 239, 382, ...
: 20, 18, 33, 71,  89, 157, 274, 419, ...
: 21, 19, 41, 86,  92, 158, 289, 457, ...
: 24, 21, 44, 87, 115, 159, 307, 458, ...
: 28, 26, 50, 98, 121, 169, 309, 571, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..200, flattened
• Wikipedia, Root of unity modulo n

Crossrefs

Columns k=1-4 give: A033949, A066498, A066500, A066502.

Row n=1 gives A066674 for k>1.

Main diagonal gives A305828.

Cf. A000010, A000040, A002322.

Programs

Maple

with(numtheory):
A:= proc() local j, l; l:= proc() [] end;
      proc(n, k)
        while nops(l(k)) lambda(j) or k>1 and
                  irem(phi(j), ithprime(k))=0 then
                  l(k):= [l(k)[], j]; break fi
          od
        od: l(k)[n]
      end
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..15);

Mathematica

A[n_, k_] := Module[{j, l = {}}, While[Length[l]CarmichaelLambda[j] || k>1 && Mod[EulerPhi[j], Prime[k]]==0, AppendTo[l, j]; Break[]]]]; l[[n]]];
Table[A[n, 1 + d - n], {d, 1, 15}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 29 2018, from Maple *)