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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).

%I A372619 #34 May 09 2024 04:49:45
%S A372619 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,
%T A372619 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,
%U A372619 5,9,10,20,19,31,33,45,42,1,2,5,7,13,12,26,23,37,37,55,46
%N 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).
%H A372619 Seiichi Manyama, <a href="/A372619/b372619.txt">Antidiagonals n = 1..140, flattened</a>
%F A372619 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
%e A372619 Square array T(n,k) begins:
%e A372619    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
%e A372619    2,  3,  2,  3,  2,  3,  2,  3,  2,  3, ...
%e A372619    4,  5,  5,  5,  4,  6,  4,  5,  5,  5, ...
%e A372619    6,  9,  7,  9,  6, 10,  6,  9,  7,  9, ...
%e A372619   10, 13, 11, 13, 11, 14, 10, 13, 11, 14, ...
%e A372619   12, 17, 14, 17, 13, 20, 12, 17, 14, 18, ...
%e A372619   18, 23, 20, 23, 19, 26, 19, 23, 20, 24, ...
%t A372619 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 *)
%o A372619 (PARI) T(n, k) = sum(j=1, n, eulerphi(k*j))/eulerphi(k);
%Y A372619 Columns k=1..10 give: A002088, A049690, A372621, A049690, A372622, A372637, A372638, A049690, A372621, A372639.
%Y A372619 Main diagonal gives A070639.
%Y A372619 Cf. A000010, A372606.
%Y A372619 Cf. A078615, A322360.
%K A372619 nonn,tabl
%O A372619 1,3
%A A372619 _Seiichi Manyama_, May 07 2024