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

1,2

Comments

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

Links

Table of n, a(n) for n=1..26.

Formula

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

Example

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

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}]

Crossrefs

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

Sequence in context: A342713 A248116 A259702 * A284923 A343850 A235707

Adjacent sequences: A377790 A377791 A377792 * A377794 A377795 A377796

Keyword

nonn,hard,more

Author

Michael De Vlieger, Nov 07 2024

Status

approved