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A350900
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Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.
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0
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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, 12, 20, 9, 20, 12, 20, 8, 21, 21, 11, 21, 21, 11, 21, 21, 9, 27, 18, 27, 18, 15, 18, 27, 18, 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
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OFFSET
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1,2
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COMMENTS
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Subtriangle (triangle without main diagonal) is symmetrical.
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).
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LINKS
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FORMULA
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T(n, k) = T(n, n-k) for 1 <= k < n.
Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.
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EXAMPLE
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The triangle T(n, k) for 1 <= k <= n starts:
n \k : 1 2 3 4 5 6 7 8 9 10 11 12
======================================================
1 : 1
2 : 3 2
3 : 5 5 3
4 : 8 5 8 4
5 : 9 9 9 9 5
6 : 15 10 9 10 15 6
7 : 13 13 13 13 13 13 7
8 : 20 12 20 9 20 12 20 8
9 : 21 21 11 21 21 11 21 21 9
10 : 27 18 27 18 15 18 27 18 27 10
11 : 21 21 21 21 21 21 21 21 21 21 11
12 : 40 25 24 20 40 15 40 20 24 25 40 12
etc.
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PROG
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(PARI) T(n, k) = sum(i=1, n, gcd(i, n) / gcd(gcd(i, k), n));
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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