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Triangle T(n,k)= tau(k)*phi(n/k) if k|n, else T(n,k)=0.
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%I #7 Aug 12 2015 21:17:28

%S 1,1,2,2,0,2,2,2,0,3,4,0,0,0,2,2,4,2,0,0,4,6,0,0,0,0,0,2,4,4,0,3,0,0,

%T 0,4,6,0,4,0,0,0,0,0,3,4,8,0,0,2,0,0,0,0,4

%N Triangle T(n,k)= tau(k)*phi(n/k) if k|n, else T(n,k)=0.

%C Tau is the number of divisors A000005, and phi the Euler totient A000010.

%F T(n,k) = A000005(k)*A054523(n,k).

%F T(n,1) = A000010(n).

%F T(n,n) = A000005(n).

%F Sum_{k=1..n} T(n,k) = A000203(n).

%e First few rows of the triangle are:

%e 1;

%e 1, 2;

%e 2, 0, 2;

%e 2, 2, 0, 3;

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

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

%e 6, 0, 0, 0, 0, 0, 2;

%e 4, 4, 0, 3, 0, 0, 0, 4;

%e ...

%p A127527 := proc(n,k) if n mod k = 0 then numtheory[tau](k)*numtheory[phi](n/k) ; else 0; end if; end proc: # _R. J. Mathar_, Apr 11 2011

%Y Cf. A054523, A000005, A000010, A000203.

%K nonn,tabl,easy

%O 1,3

%A _Gary W. Adamson_, Jan 17 2007

%E Definition clarified by _R. J. Mathar_, Apr 11 2011