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A347293
Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(1 + (i-1) * (k-1),n) for 1 <= k <= n.
0
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, 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, 27, 11, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 12, 40, 20, 24, 20, 40, 12, 40, 20, 24, 20, 40
OFFSET
1,2
COMMENTS
Triangle without column 1 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(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).
FORMULA
T(n, 1) = n; T(n, n) = A018804(n).
T(n, k) = T(n, n+2-k) for 1 < k <= n.
Conjecture: Row sums equal Dirichlet convolution of A000290 and A127473.
EXAMPLE
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 : 2 3
3 : 3 5 5
4 : 4 8 4 8
5 : 5 9 9 9 9
6 : 6 15 10 9 10 15
7 : 7 13 13 13 13 13 13
8 : 8 20 8 20 8 20 8 20
9 : 9 21 21 9 21 21 9 21 21
10 : 10 27 18 27 18 15 18 27 18 27
11 : 11 21 21 21 21 21 21 21 21 21 21
12 : 12 40 20 24 20 40 12 40 20 24 20 40
etc.
CROSSREFS
KEYWORD
nonn,easy,tabl
AUTHOR
Werner Schulte, Jan 23 2022
STATUS
approved