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

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, 35, 47, 40, 22, 7, 1, 6, 22, 53, 82, 87, 62, 29, 8, 1, 4, 28, 75, 135, 169, 149, 91, 37, 9, 1, 10, 32, 103, 210, 304, 318, 240, 128, 46, 10, 1

Offset: 1

Author

Miles Englezou, Feb 08 2024

Keywords

• nonn
• tabl

Examples

			First 10 rows and columns:
 n\k | 1   2   3    4    5     6     7      8      9     10 ...
 ----+---------------------------------------------------------
  1  | 1   1   2    2    4     2     6      4      6      4 ... = A000010
  2  | 1   2   4    6   10    12    18     22     28     32 ... = A002088
  3  | 1   3   7   13   23    35    53     75    103    135 ... = A103116
  4  | 1   4  11   24   47    82   135    210    313    448 ...
  5  | 1   5  16   40   87   169   304    514    827   1275 ...
  6  | 1   6  22   62  149   318   622   1136   1963   3238 ...
  7  | 1   7  29   91  240   558  1180   2316   4279   7517 ...
  8  | 1   8  37  128  368   926  2106   4422   8701  16218 ...
  9  | 1   9  46  174  542  1468  3574   7996  16697  32915 ...
 10  | 1  10  56  230  772  2240  5814  13810  30507  63422 ...
 ...
		

Crossrefs

Cf. A000010 (Euler phi), A002088 (Euler phi partial sums), A103116 (Euler phi partial sums two iterations).

Programs

MATLAB

function out = a(n)
    Z = zeros(n);
    A = arrayfun(@eulerPhi,[1:n]);
    Z(1,1:n) = A;
    for i = 2 : n
        A = cumsum(A);
        Z(i,1:n) = A;
    end
    [nr,nc] = size(Z);
    [R,C] = ndgrid(1:nr,1:nc);
    M = [reshape(R+C,[],1),R(:)];
    [~,ind] = sortrows(M);
    Z = Z(ind)';
    out = Z(1,n);

Mathematica

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 *)

Formula

T(1,k) = A000010(k) for k >= 1; T(n,k) = Sum_{i=1..k} T(n-1,i) for n > 1, k >= 1.