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

Offset: 1

Author

Werner Schulte, Jan 23 2022

Keywords

• nonn
• easy
• tabl

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).

Examples

			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

Cf. A000010, A000290, A008683, A018804, A127473.

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.