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

1,1

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.

Links

Alois P. Heinz, Antidiagonals n = 1..200, flattened

Wikipedia, Root of unity modulo n

Example

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, ...

Maple

with(numtheory):
A:= proc() local j, l; l:= proc() [] end;
      proc(n, k)
        while nops(l(k))<n do
          for j from 1+`if`(l(k)=[], 0, l(k)[-1]) do
            if k=1 and phi(j)> 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]<n, For[j = 1+If[l == {}, 0, l[[-1]]], True, j++, If[k==1 && EulerPhi[j]>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 *)

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.

Sequence in context: A212911 A019326 A286839 * A051011 A298666 A309628

Adjacent sequences: A277912 A277913 A277914 * A277916 A277917 A277918

Keyword

nonn,look,tabl

Author

Alois P. Heinz, Nov 03 2016

Status

approved