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%I #13 May 24 2024 11:58:43
%S 1,3,2,5,5,3,8,5,8,4,9,9,9,9,5,15,10,9,10,15,6,13,13,13,13,13,13,7,20,
%T 12,20,9,20,12,20,8,21,21,11,21,21,11,21,21,9,27,18,27,18,15,18,27,18,
%U 27,10,21,21,21,21,21,21,21,21,21,21,11,40,25,24,20,40,15,40,20,24,25,40,12
%N Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.
%C Subtriangle (triangle without main diagonal) is symmetrical.
%C Conjecture: Let f be an arbitrary arithmetic function. Define for n > 0 the sequence a(f; n) = Sum_{i=1..n, k=1..n} f(gcd(i,n)/gcd(gcd(i,k),n)); a(f; n) equals Dirichlet convolution of f(n)*A000010(n) and A057660(n); if f is multiplicative, then a(f; n) is multiplicative; row sums of this triangle use f(n) = n (see formula section).
%F T(n, 1) = A018804(n); T(n, n) = n.
%F T(n, k) = T(n, n-k) for 1 <= k < n.
%F Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.
%e The triangle T(n, k) for 1 <= k <= n starts:
%e n \k : 1 2 3 4 5 6 7 8 9 10 11 12
%e ======================================================
%e 1 : 1
%e 2 : 3 2
%e 3 : 5 5 3
%e 4 : 8 5 8 4
%e 5 : 9 9 9 9 5
%e 6 : 15 10 9 10 15 6
%e 7 : 13 13 13 13 13 13 7
%e 8 : 20 12 20 9 20 12 20 8
%e 9 : 21 21 11 21 21 11 21 21 9
%e 10 : 27 18 27 18 15 18 27 18 27 10
%e 11 : 21 21 21 21 21 21 21 21 21 21 11
%e 12 : 40 25 24 20 40 15 40 20 24 25 40 12
%e etc.
%o (PARI) T(n, k) = sum(i=1, n, gcd(i,n) / gcd(gcd(i,k),n));
%o row(n) = vector(n, k, T(n,k)); \\ _Michel Marcus_, Jan 22 2022
%Y Row sums gives A373059.
%Y Cf. A000010, A002618, A018804, A057660, A050873.
%K nonn,easy,tabl,changed
%O 1,2
%A _Werner Schulte_, Jan 21 2022
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