A372606 - OEIS

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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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

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

EXAMPLE

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, ...

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 *)

PROG