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A370075 Iterated partial sums of Euler totient function (A000010). Square array read by descending antidiagonals.

%I A370075 #14 Feb 26 2024 10:42:53
%S A370075 1,1,1,2,2,1,2,4,3,1,4,6,7,4,1,2,10,13,11,5,1,6,12,23,24,16,6,1,4,18,
%T A370075 35,47,40,22,7,1,6,22,53,82,87,62,29,8,1,4,28,75,135,169,149,91,37,9,
%U A370075 1,10,32,103,210,304,318,240,128,46,10,1
%N A370075 Iterated partial sums of Euler totient function (A000010). Square array read by descending antidiagonals.
%F A370075 T(1,k) = A000010(k) for k >= 1; T(n,k) = Sum_{i=1..k} T(n-1,i) for n > 1, k >= 1.
%e A370075 First 10 rows and columns:
%e A370075  n\k | 1   2   3    4    5     6     7      8      9     10 ...
%e A370075  ----+---------------------------------------------------------
%e A370075   1  | 1   1   2    2    4     2     6      4      6      4 ... = A000010
%e A370075   2  | 1   2   4    6   10    12    18     22     28     32 ... = A002088
%e A370075   3  | 1   3   7   13   23    35    53     75    103    135 ... = A103116
%e A370075   4  | 1   4  11   24   47    82   135    210    313    448 ...
%e A370075   5  | 1   5  16   40   87   169   304    514    827   1275 ...
%e A370075   6  | 1   6  22   62  149   318   622   1136   1963   3238 ...
%e A370075   7  | 1   7  29   91  240   558  1180   2316   4279   7517 ...
%e A370075   8  | 1   8  37  128  368   926  2106   4422   8701  16218 ...
%e A370075   9  | 1   9  46  174  542  1468  3574   7996  16697  32915 ...
%e A370075  10  | 1  10  56  230  772  2240  5814  13810  30507  63422 ...
%e A370075  ...
%t A370075 T[1, k_] := EulerPhi[k]; T[n_, k_] := T[n, k] = Sum[T[n - 1, i], {i, 1, k}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Feb 09 2024 *)
%o A370075 (MATLAB)
%o A370075 function out = a(n)
%o A370075     Z = zeros(n);
%o A370075     A = arrayfun(@eulerPhi,[1:n]);
%o A370075     Z(1,1:n) = A;
%o A370075     for i = 2 : n
%o A370075         A = cumsum(A);
%o A370075         Z(i,1:n) = A;
%o A370075     end
%o A370075     [nr,nc] = size(Z);
%o A370075     [R,C] = ndgrid(1:nr,1:nc);
%o A370075     M = [reshape(R+C,[],1),R(:)];
%o A370075     [~,ind] = sortrows(M);
%o A370075     Z = Z(ind)';
%o A370075     out = Z(1,n);
%Y A370075 Cf. A000010 (Euler phi), A002088 (Euler phi partial sums), A103116 (Euler phi partial sums two iterations).
%K A370075 nonn,tabl
%O A370075 1,4
%A A370075 _Miles Englezou_, Feb 08 2024