A347293 - OEIS

A347293

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

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

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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).

LINKS

Table of n, a(n) for n=1..78.

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

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1 : 1

2 : 2 3