A186230 - OEIS

%I #32 Sep 21 2018 22:11:22

%S 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,

%T 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,

%U 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

%N 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.

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

%H Alois P. Heinz, <a href="/A186230/b186230.txt">Rows n = 1..250, flattened</a>

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

%e T(n,1) = 0 because no positive integer j<1 can be found.

%e T(n,k) = 0 if GCD(n,k)>1.

%e T(7,5) = 4 because for j in {1,2,3,4} all conditions are satisfied.

%e Triangle T(n,k) begins:

%e   0;

%e   0, 0;

%e   0, 1, 0;

%e   0, 0, 1, 0;

%e   0, 1, 2, 2, 0;

%e   0, 0, 0, 0, 1, 0;

%e   0, 1, 2, 2, 4, 2, 0;

%p with(numtheory):

%p T:= proc(n,k) local c, i, j, m;

%p       if k=1 or igcd(n, k)>1 then 0

%p     elif isprime(n) then phi(k)

%p     else m:= n*k;

%p          i:= igcd(m, 2);

%p          c:= 0;

%p          for j to k-1 by i do

%p            if igcd(m, j)=1 then c:= c+1 fi

%p          od; c

%p       fi

%p     end:

%p seq(seq(T(n, k), k=1..n), n=1..20);

%t 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 *)

%Y 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.

%K nonn,tabl,look

%O 1,13

%A _Alois P. Heinz_, Feb 15 2011