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A350900 Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.

%I A350900 #13 May 24 2024 11:58:43
%S A350900 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 A350900 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 A350900 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 A350900 Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.
%C A350900 Subtriangle (triangle without main diagonal) is symmetrical.
%C A350900 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 A350900 T(n, 1) = A018804(n); T(n, n) = n.
%F A350900 T(n, k) = T(n, n-k) for 1 <= k < n.
%F A350900 Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.
%e A350900 The triangle T(n, k) for 1 <= k <= n starts:
%e A350900 n \k :   1   2   3   4   5   6   7   8   9  10  11  12
%e A350900 ======================================================
%e A350900    1 :   1
%e A350900    2 :   3   2
%e A350900    3 :   5   5   3
%e A350900    4 :   8   5   8   4
%e A350900    5 :   9   9   9   9   5
%e A350900    6 :  15  10   9  10  15   6
%e A350900    7 :  13  13  13  13  13  13   7
%e A350900    8 :  20  12  20   9  20  12  20   8
%e A350900    9 :  21  21  11  21  21  11  21  21   9
%e A350900   10 :  27  18  27  18  15  18  27  18  27  10
%e A350900   11 :  21  21  21  21  21  21  21  21  21  21  11
%e A350900   12 :  40  25  24  20  40  15  40  20  24  25  40  12
%e A350900   etc.
%o A350900 (PARI) T(n, k) = sum(i=1, n, gcd(i,n) / gcd(gcd(i,k),n));
%o A350900 row(n) = vector(n, k, T(n,k)); \\ _Michel Marcus_, Jan 22 2022
%Y A350900 Row sums gives A373059.
%Y A350900 Cf. A000010, A002618, A018804, A057660, A050873.
%K A350900 nonn,easy,tabl
%O A350900 1,2
%A A350900 _Werner Schulte_, Jan 21 2022