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Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} phi(k*j).
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%I #36 May 10 2024 03:34:36

%S 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,

%T 12,12,24,26,28,23,22,6,12,24,20,44,34,40,31,28,4,12,20,36,28,52,46,

%U 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

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

%H Seiichi Manyama, <a href="/A372606/b372606.txt">Antidiagonals n = 1..140, flattened</a>

%F 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

%e Square array T(n,k) begins:

%e 1, 1, 2, 2, 4, 2, 6, ...

%e 2, 3, 4, 6, 8, 6, 12, ...

%e 4, 5, 10, 10, 16, 12, 24, ...

%e 6, 9, 14, 18, 24, 20, 36, ...

%e 10, 13, 22, 26, 44, 28, 60, ...

%e 12, 17, 28, 34, 52, 40, 72, ...

%e 18, 23, 40, 46, 76, 52, 114, ...

%t 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 *)

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

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

%Y Main diagonal gives A372608.

%Y Cf. A000010, A372619.

%Y Cf. A007947, A048250, A332880, A332881.

%K nonn,tabl

%O 1,3

%A _Seiichi Manyama_, May 07 2024