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A370454 a(n) = 1 + ceiling((log q)/(log p)), where p = A020639(s) and q = A119288(s) is the second smallest distinct prime factor of squarefree composite s = A120944(n).

%I A370454 #5 Feb 25 2024 12:43:21
%S A370454 3,4,4,3,3,5,5,3,4,6,3,6,4,3,6,4,3,4,6,6,3,3,4,4,7,3,3,7,3,7,5,3,5,7,
%T A370454 3,3,3,7,4,5,3,3,7,3,7,5,5,4,3,8,3,5,8,3,4,8,4,4,8,5,3,3,8,4,3,5,8,4,
%U A370454 5,4,3,3,4,8,3,5,8,3,4,8,3,3,5,8,4,3,8
%N A370454 a(n) = 1 + ceiling((log q)/(log p)), where p = A020639(s) and q = A119288(s) is the second smallest distinct prime factor of squarefree composite s = A120944(n).
%H A370454 Michael De Vlieger, <a href="/A370454/b370454.txt">Table of n, a(n) for n = 1..10000</a>
%F A370454 Let c(s) be the number of powers p^m of p = lpf(s) = A020639(s) that precede q = A119288(s) in the sequence { k = m*s : rad(m) | s }, where rad(n) = A007947(n).
%F A370454 a(n) = 1 + c(A120944(n)).
%e A370454 Let b(n) = A120944(n).
%e A370454 a(1) = 3 since b(1) = 6, p = 2, and q = 3; 1 + Ceiling(log 3/log 2) = 3.
%e A370454   For s = 6, { k = m*s : rad(m) | s } = A003586 begins {1, 2, 3, 4, 6, ...};
%e A370454   there are 2 powers of 2 before q = 3 so c(6) = 2 = a(1) - 1.
%e A370454 a(2) = 4 since b(2) = 10, p = 2, and q = 5; 1 + Ceiling(log 5/log 2) = 4.
%e A370454   For s = 10, { k = m*s : rad(m) | s } = A003592 begins {1, 2, 4, 5, 8, 10, ...};
%e A370454   there are 3 powers of 2 before q = 5 so c(10) = 3 = a(2) - 1.
%e A370454 a(6) = 5 since b(6) = 22, p = 2, and q = 11; 1 + Ceiling(log 11/log 2) = 5.
%e A370454   For s = 22, { k = m*s : rad(m) | s } = A003596 begins {1, 2, 4, 8, 11, ...};
%e A370454   there are 4 powers of 2 before q = 11 so c(22) = 4 = a(6) - 1, etc
%t A370454 Map[1 + Ceiling[Log[##]] & @@ FactorInteger[#][[1 ;; 2, 1]] &, Select[Range[300], And[CompositeQ[#], SquareFreeQ[#]] &]]
%Y A370454 Cf. A007947, A020639, A119288, A120944.
%K A370454 nonn,easy
%O A370454 1,1
%A A370454 _Michael De Vlieger_, Feb 18 2024