A332977 Triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m satisfying Omega(m) + pi(gpf(m)) - [m<>1] = n; n>=0, 1<=k<=A011782(n).

1, 2, 3, 4, 5, 6, 8, 9, 7, 10, 12, 15, 16, 18, 25, 27, 11, 14, 20, 21, 24, 30, 32, 35, 36, 45, 49, 50, 54, 75, 81, 125, 13, 22, 28, 33, 40, 42, 48, 55, 60, 63, 64, 70, 72, 77, 90, 98, 100, 105, 108, 121, 135, 147, 150, 162, 175, 225, 243, 245, 250, 343, 375, 625

Offset: 0

Author

Alois P. Heinz, Mar 04 2020

Keywords

• nonn
• tabf

Comments

Integer m > 0 is listed in row n if the index of the largest prime factor of m (or 0 for empty prime factor set) plus the cardinality of the other prime factors of m (counted with multiplicity) equals n.

Row n+k-1 contains prime(n)^k (for all n, k >= 1).

The concatenation of all rows (with offset 1) gives a permutation of the natural numbers A000027 with fixed points 1, 2, 3, 4, 5, 6, 10, ... and inverse permutation A332990.

This is a variant with sorted rows of A005940 (offset differs) or A163511.

Examples

			Triangle T(n,k) begins:
   1;
   2;
   3,  4;
   5,  6,  8,  9;
   7, 10, 12, 15, 16, 18, 25, 27;
  11, 14, 20, 21, 24, 30, 32, 35, 36, 45, 49, 50, 54, 75, 81, 125;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..16, flattened
• Wikipedia, Iverson bracket
• Index entries for sequences that are permutations of the natural numbers
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Columns k=1-2 give: A008578(n+1), A100484(n-1) for n>1.

Last elements of rows give A332979.

Row sums give A252737.

Product of row elements give A252738.

Row lengths give A011782.

Cf. A000027, A000040, A000720 (pi), A001222 (Omega), A006530 (GPF), A060576 ([n<>1]), A061395 (pi(gpf(n))), A215366, A332990.

Cf. A005940, A163511.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1], sort([seq(map(x-> x*
      ithprime(j), b(n-`if`(i=0, j, 1), j))[], j=1..`if`(i=0, n, i))]))
    end:
T:= n-> b(n, 0)[]:
seq(T(n), n=0..7);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {1}, Sort[Flatten[Table[#*
    Prime[j]& /@ b[n-If[i == 0, j, 1], j], {j, 1, If[i == 0, n, i]}]]]];
T[n_] := b[n, 0];
T /@ Range[0, 7] // Flatten (* Jean-François Alcover, Mar 30 2021, after Alois P. Heinz *)