A186230 Triangle T(n,k), n>=1, 1<=k<=n, read by rows: T(n,k) is the number of positive integers j<k such that j,k,n are pairwise coprime.

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

1,13

Comments

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

Links

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

Formula

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

Example

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;

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 *)

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.

Sequence in context: A359887 A033148 A281084 * A214304 A248640 A281083

Adjacent sequences: A186227 A186228 A186229 * A186231 A186232 A186233

Keyword

nonn,tabl,look

Author

Alois P. Heinz, Feb 15 2011

Status

approved