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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

Table of n, a(n) for n=1..123.

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

Cf. A033950, A208251, A000720.

Sequence in context: A255010 A292371 A216683 * A274884 A076224 A286582

Adjacent sequences:  A323323 A323324 A323325 * A323327 A323328 A323329

KEYWORD

nonn

AUTHOR

Jud McCranie, Jan 11 2019

STATUS

approved

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Last modified February 22 15:03 EST 2020. Contains 332137 sequences. (Running on oeis4.)