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).

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, 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, 5, 4, 3, 3, 4, 8, 3, 5, 8, 3, 4, 8, 3, 3, 5, 8, 4, 3, 8

Offset: 1

Author

Michael De Vlieger, Feb 18 2024

Keywords

• nonn
• easy

Examples

			Let b(n) = A120944(n).
a(1) = 3 since b(1) = 6, p = 2, and q = 3; 1 + Ceiling(log 3/log 2) = 3.
  For s = 6, { k = m*s : rad(m) | s } = A003586 begins {1, 2, 3, 4, 6, ...};
  there are 2 powers of 2 before q = 3 so c(6) = 2 = a(1) - 1.
a(2) = 4 since b(2) = 10, p = 2, and q = 5; 1 + Ceiling(log 5/log 2) = 4.
  For s = 10, { k = m*s : rad(m) | s } = A003592 begins {1, 2, 4, 5, 8, 10, ...};
  there are 3 powers of 2 before q = 5 so c(10) = 3 = a(2) - 1.
a(6) = 5 since b(6) = 22, p = 2, and q = 11; 1 + Ceiling(log 11/log 2) = 5.
  For s = 22, { k = m*s : rad(m) | s } = A003596 begins {1, 2, 4, 8, 11, ...};
  there are 4 powers of 2 before q = 11 so c(22) = 4 = a(6) - 1, etc
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007947, A020639, A119288, A120944.

Programs

Mathematica

Map[1 + Ceiling[Log[##]] & @@ FactorInteger[#][[1 ;; 2, 1]] &, Select[Range[300], And[CompositeQ[#], SquareFreeQ[#]] &]]

Formula

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).

a(n) = 1 + c(A120944(n)).