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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).

%I A372606 #36 May 10 2024 03:34:36
%S A372606 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 A372606 12,12,24,26,28,23,22,6,12,24,20,44,34,40,31,28,4,12,20,36,28,52,46,
%U A372606 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 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).
%H A372606 Seiichi Manyama, <a href="/A372606/b372606.txt">Antidiagonals n = 1..140, flattened</a>
%F A372606 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 A372606 Square array T(n,k) begins:
%e A372606    1,  1,  2,  2,  4,  2,   6, ...
%e A372606    2,  3,  4,  6,  8,  6,  12, ...
%e A372606    4,  5, 10, 10, 16, 12,  24, ...
%e A372606    6,  9, 14, 18, 24, 20,  36, ...
%e A372606   10, 13, 22, 26, 44, 28,  60, ...
%e A372606   12, 17, 28, 34, 52, 40,  72, ...
%e A372606   18, 23, 40, 46, 76, 52, 114, ...
%t A372606 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 A372606 (PARI) T(n, k) = sum(j=1, n, eulerphi(k*j));
%Y A372606 Columns k=1..2 give: A002088, A049690.
%Y A372606 Main diagonal gives A372608.
%Y A372606 Cf. A000010, A372619.
%Y A372606 Cf. A007947, A048250, A332880, A332881.
%K A372606 nonn,tabl
%O A372606 1,3
%A A372606 _Seiichi Manyama_, May 07 2024