A378638 Irregular triangle read by rows: row n lists all m such that phi(m) divides n, where phi is the Euler totient function (A000010).

1, 2, 1, 2, 3, 4, 6, 1, 2, 1, 2, 3, 4, 5, 6, 8, 10, 12, 1, 2, 1, 2, 3, 4, 6, 7, 9, 14, 18, 1, 2, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 1, 2, 1, 2, 3, 4, 6, 11, 22, 1, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 18, 21, 26, 28, 36, 42, 1, 2, 1, 2, 3, 4, 6, 1, 2

Offset

1,2

Comments

If n is odd, row n is {1, 2}.

If n is even, row n begins with {1, 2, 3, 4}.

Links

Paolo Xausa, Table of n, a(n) for n = 1..13988 (rows 1..1000 of triangle, flattened).

Formula

T(n,k) <= n^2, for n > 2 (see A319048).

Example

Triangle begins:
  n\k| 1  2  3  4  5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20 ...
  ----------------------------------------------------------------------------------
   1 | 1, 2;
   2 | 1, 2, 3, 4, 6;
   3 | 1, 2;
   4 | 1, 2, 3, 4, 5,  6,  8, 10, 12;
   5 | 1, 2;
   6 | 1, 2, 3, 4, 6,  7,  9, 14, 18;
   7 | 1, 2;
   8 | 1, 2, 3, 4, 5,  6,  8, 10, 12, 15, 16, 20, 24, 30;
   9 | 1, 2;
  10 | 1, 2, 3, 4, 6, 11, 22;
  11 | 1, 2;
  12 | 1, 2, 3, 4, 5,  6,  7,  8,  9, 10, 12, 13, 14, 18, 21, 26, 28, 36, 42;
  13 | 1, 2;
  14 | 1, 2, 3, 4, 6;
  15 | 1, 2;
  16 | 1, 2, 3, 4, 5,  6,  8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 60;
  17 | 1, 2;
  18 | 1, 2, 3, 4, 6,  7,  9, 14, 18, 19, 27, 38, 54;
  19 | 1, 2;
  20 | 1, 2, 3, 4, 5,  6,  8, 10, 11, 12, 22, 25, 33, 44, 50, 66;
  ...

Mathematica

With[{nmax = 25}, Table[If[OddQ[n], {1, 2}, PositionIndex[Divisible[n, #[[;; Max[n^2, 6]]]]][True]], {n, nmax}] & [EulerPhi[Range[nmax^2]]]]

Prog

(PARI) row(n) = select(x->!(n % eulerphi(x)), [1..max(n^2, 6)]); \\ Michel Marcus, Dec 05 2024

Crossrefs

Cf. A070633 (row lengths), A319048 (right border), A378639 (row sums).

Supersequence of A378636.

Cf. A000010.

Sequence in context: A220237 A050040 A277282 * A191973 A290973 A173497

Adjacent sequences: A378634 A378636 A378637 * A378639 A378640 A378641

Keyword

nonn,tabf,new

Author

Paolo Xausa, Dec 03 2024

Status

approved