A186230 Triangle T(n,k), n>=1, 1<=k<=n, read by rows: T(n,k) is the number of positive integers j

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 2, 4, 2, 0, 0, 0, 1, 0, 2, 0, 3, 0, 0, 1, 0, 1, 3, 0, 4, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 5, 0

Offset: 1

Author

Alois P. Heinz, Feb 15 2011

Keywords

• nonn
• tabl
• look

Comments

T(n,k) = A000010(k) if n is prime and 1

Examples

			T(n,1) = 0 because no positive integer j<1 can be found.
T(n,k) = 0 if GCD(n,k)>1.
T(7,5) = 4 because for j in {1,2,3,4} all conditions are satisfied.
Triangle T(n,k) begins:
  0;
  0, 0;
  0, 1, 0;
  0, 0, 1, 0;
  0, 1, 2, 2, 0;
  0, 0, 0, 0, 1, 0;
  0, 1, 2, 2, 4, 2, 0;
		

Links

• Alois P. Heinz, Rows n = 1..250, flattened

Crossrefs

Row sums give: A185953. Column k=2 gives: A000035 for n>1. Lower diagonal gives: A057475(n-1) for n>2. Cf. A000010, A000040, A003989.

Programs

Maple

with(numtheory):
T:= proc(n,k) local c, i, j, m;
      if k=1 or igcd(n, k)>1 then 0
    elif isprime(n) then phi(k)
    else m:= n*k;
         i:= igcd(m, 2);
         c:= 0;
         for j to k-1 by i do
           if igcd(m, j)=1 then c:= c+1 fi
         od; c
      fi
    end:
seq(seq(T(n, k), k=1..n), n=1..20);

Mathematica

t[n_, k_] := Module[{c, i, j, m}, If[ k == 1 || GCD[n, k] > 1, 0, If[PrimeQ[n], EulerPhi[k], m = n*k; i = GCD[m, 2]; c = 0; For[j = 1, j <= k-1, j = j+i, If[GCD[m, j] == 1, c = c+1]]; c]]]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)

Formula

T(n,k) = |{ j : 1 <= j < k and GCD(n,k) = GCD(n,j) = GCD(k,j) = 1 }|.

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