A372619 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = 1/(phi(k)) * Sum_{j=1..n} phi(k*j).

1, 1, 2, 1, 3, 4, 1, 2, 5, 6, 1, 3, 5, 9, 10, 1, 2, 5, 7, 13, 12, 1, 3, 4, 9, 11, 17, 18, 1, 2, 6, 6, 13, 14, 23, 22, 1, 3, 4, 10, 11, 17, 20, 31, 28, 1, 2, 5, 6, 14, 13, 23, 24, 37, 32, 1, 3, 5, 9, 10, 20, 19, 31, 33, 45, 42, 1, 2, 5, 7, 13, 12, 26, 23, 37, 37, 55, 46

Offset: 1

Author

Seiichi Manyama, May 07 2024

Keywords

• nonn
• tabl

Examples

			Square array T(n,k) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   2,  3,  2,  3,  2,  3,  2,  3,  2,  3, ...
   4,  5,  5,  5,  4,  6,  4,  5,  5,  5, ...
   6,  9,  7,  9,  6, 10,  6,  9,  7,  9, ...
  10, 13, 11, 13, 11, 14, 10, 13, 11, 14, ...
  12, 17, 14, 17, 13, 20, 12, 17, 14, 18, ...
  18, 23, 20, 23, 19, 26, 19, 23, 20, 24, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 1..140, flattened

Crossrefs

Columns k=1..10 give: A002088, A049690, A372621, A049690, A372622, A372637, A372638, A049690, A372621, A372639.

Main diagonal gives A070639.

Cf. A000010, A372606.

Cf. A078615, A322360.

Programs

Mathematica

T[n_, k_] := Sum[EulerPhi[k*j], {j, 1, n}] / EulerPhi[k]; Table[T[k, n-k+1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2024 *)

PARI

T(n, k) = sum(j=1, n, eulerphi(k*j))/eulerphi(k);

Formula

T(n,k) ~ (3/Pi^2) * c(k) * n^2, where c(k) = A078615(k)/A322360(k) is the multiplicative function defined by c(p^e) = p^2/(p^2-1). - Amiram Eldar, May 09 2024