A323326 a(n) = 2*T(n) - pi(n), where T(n) (A208251) is the number of refactorable/tau numbers (A033950) <= n and pi(n) (A000720) is the number of primes <= n.

2, 3, 2, 2, 1, 1, 0, 2, 4, 4, 3, 5, 4, 4, 4, 4, 3, 5, 4, 4, 4, 4, 3, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 5, 4, 4, 4, 6, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 4, 4, 4, 3, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 2, 4, 4, 4, 3, 5, 5, 5, 5, 7, 6, 6, 6, 6, 6, 6, 6, 8, 7, 7, 7, 7, 6, 6, 5, 7, 7, 7, 6, 8, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Author

Jud McCranie, Jan 11 2019

Keywords

• nonn

Comments

Colton conjectured that T(n) >= pi(n)/2 for all n, i.e., this sequence is nonnegative. Zelinsky proved it for n > 7.42*10^13 (see the Zelinsky reference). This calculation went to 7.44*10^13, proving the conjecture.

Examples

			For n=6, pi(6)=3, T(6)=2, so a(6) = 2*2 - 3 = 1.
		

Links

• Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999.
• Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Crossrefs

Cf. A033950, A208251, A000720.