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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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
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.
EXAMPLE
For n=6, pi(6)=3, T(6)=2, so a(6) = 2*2 - 3 = 1.
CROSSREFS
Sequence in context: A368123 A368128 A362956 * A274884 A076224 A286582
KEYWORD
nonn
AUTHOR
Jud McCranie, Jan 11 2019
STATUS
approved

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Last modified June 23 15:10 EDT 2024. Contains 373651 sequences. (Running on oeis4.)