A372606 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} phi(k*j).

1, 1, 2, 2, 3, 4, 2, 4, 5, 6, 4, 6, 10, 9, 10, 2, 8, 10, 14, 13, 12, 6, 6, 16, 18, 22, 17, 18, 4, 12, 12, 24, 26, 28, 23, 22, 6, 12, 24, 20, 44, 34, 40, 31, 28, 4, 12, 20, 36, 28, 52, 46, 48, 37, 32, 10, 12, 30, 36, 60, 40, 76, 62, 66, 45, 42, 4, 20, 20, 42, 52, 72, 52, 92, 74, 74, 55, 46

Offset: 1

Author

Seiichi Manyama, May 07 2024

Keywords

• nonn
• tabl

Examples

			Square array T(n,k) begins:
   1,  1,  2,  2,  4,  2,   6, ...
   2,  3,  4,  6,  8,  6,  12, ...
   4,  5, 10, 10, 16, 12,  24, ...
   6,  9, 14, 18, 24, 20,  36, ...
  10, 13, 22, 26, 44, 28,  60, ...
  12, 17, 28, 34, 52, 40,  72, ...
  18, 23, 40, 46, 76, 52, 114, ...
		

Links

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

Crossrefs

Columns k=1..2 give: A002088, A049690.

Main diagonal gives A372608.

Cf. A000010, A372619.

Cf. A007947, A048250, A332880, A332881.

Programs

Mathematica

T[n_, k_] := Sum[EulerPhi[k*j], {j, 1, n}]; Table[T[k, n-k+1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 10 2024 *)

PARI

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

Formula

T(n,k) ~ (3/Pi^2) * c(k) * n^2, where c(k) = k * A007947(k)/A048250(k) = k * A332881(k) / A332880(k) is the multiplicative function defined by c(p^e) = p^(e+1)/(p+1). - Amiram Eldar, May 10 2024