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%I #23 Jan 26 2022 08:58:45
%S 1,2,3,3,5,5,4,8,4,8,5,9,9,9,9,6,15,10,9,10,15,7,13,13,13,13,13,13,8,
%T 20,8,20,8,20,8,20,9,21,21,9,21,21,9,21,21,10,27,18,27,18,15,18,27,18,
%U 27,11,21,21,21,21,21,21,21,21,21,21,12,40,20,24,20,40,12,40,20,24,20,40
%N Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(1 + (i-1) * (k-1),n) for 1 <= k <= n.
%C Triangle without column 1 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(1 + (i-1) * (k-1),n)); then a(f; n) = dc(A000290(n), A000010(n) * dc(A008683(n), f(n)) where dc(x, y) is Dirichlet convolution of x and y; 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) = n; T(n, n) = A018804(n).
%F T(n, k) = T(n, n+2-k) for 1 < k <= n.
%F Conjecture: Row sums equal Dirichlet convolution of A000290 and A127473.
%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 : 2 3
%e 3 : 3 5 5
%e 4 : 4 8 4 8
%e 5 : 5 9 9 9 9
%e 6 : 6 15 10 9 10 15
%e 7 : 7 13 13 13 13 13 13
%e 8 : 8 20 8 20 8 20 8 20
%e 9 : 9 21 21 9 21 21 9 21 21
%e 10 : 10 27 18 27 18 15 18 27 18 27
%e 11 : 11 21 21 21 21 21 21 21 21 21 21
%e 12 : 12 40 20 24 20 40 12 40 20 24 20 40
%e etc.
%Y Cf. A000010, A000290, A008683, A018804, A127473.
%K nonn,easy,tabl
%O 1,2
%A _Werner Schulte_, Jan 23 2022
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