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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.

%I A377793 #7 Jun 02 2025 15:28:29
%S A377793 1,2,9,21,128,194,713,874,2276,11898,12522,52469,103824,99930,173685,
%T A377793 534743,1608864,1438340,3894769,5881191,5008669,11802600,16274460,
%U A377793 36220208,132526590,178177142
%N 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.
%C A377793 a(n) is the number of terms in A377713 with least prime factor prime(n).
%F A377793 a(n) = length of row n of A377792.
%e A377793 In A377713, there are terms k with smallest prime factor prime(n) as follows:
%e A377793     Prime(n)  | a(n) | k such that floor(log_lpf(k) k) <= Omega(k)
%e A377793 -------------------------------------------------------------------------------
%e A377793 prime(1) =  2 |   1  | 6
%e A377793 prime(2) =  3 |   2  | 15, 27
%e A377793 prime(3) =  5 |   9  | 35, 55, 65, 85, 95, 115, 385, 455, 595
%e A377793 prime(4) =  7 |  21  | 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 1001,
%e A377793               |      | 1309, 1463, 1547, 1729, 1771, 2093, 2233, 2261, 2387
%e A377793 prime(5) = 11 | 128  | 143, 187, 209, ..., 1733303
%t A377793 Table[c = 0; p = Prime[i]; m = p^3;
%t A377793   Set[{w, t}, {{p, NextPrime[p]}, False}];
%t A377793   Do[Set[s, Times @@ w];
%t A377793     If[s < m,
%t A377793       AppendTo[w, NextPrime@ Last[w] ]; m *= p; c++,
%t A377793       If[Length[w] < 3, Break[],
%t A377793         w = Append[w[[;; -3]], NextPrime@ w[[-2]] ]; m /= p] ],
%t A377793     Infinity]; c, {i, 12}]
%Y A377793 Cf. A001222, A020639, A120944, A377713, A377792.
%K A377793 nonn,hard,more
%O A377793 1,2
%A A377793 _Michael De Vlieger_, Nov 07 2024