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

1,4

Links

Table of n, a(n) for n=1..66.

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.

Example

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 ...
 ...

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 *)

Prog

(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);

Crossrefs

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

Sequence in context: A274742 A128176 A368415 * A144963 A305632 A329069

Adjacent sequences: A370072 A370073 A370074 * A370076 A370077 A370078

Keyword

nonn,tabl

Author

Miles Englezou, Feb 08 2024

Status

approved