A377793 a(n) is the number of squarefree composite k with lpf(k) = prime(n) such that m <= Omega(k), where lpf = A020639, m = floor(log k / log lpf(k)), and Omega = A001222.

1, 2, 9, 21, 128, 194, 713, 874, 2276, 11898, 12522, 52469, 103824, 99930, 173685, 534743, 1608864, 1438340, 3894769, 5881191, 5008669, 11802600, 16274460, 36220208, 132526590, 178177142

Offset: 1

Author

Michael De Vlieger, Nov 07 2024

Keywords

• nonn
• hard
• more

Comments

a(n) is the number of terms in A377713 with least prime factor prime(n).

Examples

			In A377713, there are terms k with smallest prime factor prime(n) as follows:
    Prime(n)  | a(n) | k such that floor(log_lpf(k) k) <= Omega(k)
-------------------------------------------------------------------------------
prime(1) =  2 |   1  | 6
prime(2) =  3 |   2  | 15, 27
prime(3) =  5 |   9  | 35, 55, 65, 85, 95, 115, 385, 455, 595
prime(4) =  7 |  21  | 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 1001,
              |      | 1309, 1463, 1547, 1729, 1771, 2093, 2233, 2261, 2387
prime(5) = 11 | 128  | 143, 187, 209, ..., 1733303
		

Crossrefs

Cf. A001222, A020639, A120944, A377713, A377792.

Programs

Mathematica

Table[c = 0; p = Prime[i]; m = p^3;
  Set[{w, t}, {{p, NextPrime[p]}, False}];
  Do[Set[s, Times @@ w];
    If[s < m,
      AppendTo[w, NextPrime@ Last[w] ]; m *= p; c++,
      If[Length[w] < 3, Break[],
        w = Append[w[[;; -3]], NextPrime@ w[[-2]] ]; m /= p] ],
    Infinity]; c, {i, 12}]

Formula

a(n) = length of row n of A377792.