A069754 Counts transitions between prime and nonprime to reach the number n.

0, 1, 1, 2, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 10, 10, 11, 12, 13, 14, 14, 14, 15, 16, 16, 16, 16, 16, 17, 18, 19, 20, 20, 20, 20, 20, 21, 22, 22, 22, 23, 24, 25, 26, 26, 26, 27, 28, 28, 28, 28, 28, 29, 30, 30, 30, 30, 30, 31, 32, 33, 34, 34, 34, 34, 34, 35, 36, 36, 36, 37, 38, 39

Offset: 1

Author

T. D. Noe, May 02 2002

Keywords

• easy
• nice
• nonn

Comments

The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002

Examples

			a(6) = 4 because there are 4 transitions: 1 to 2, 3 to 4, 4 to 5 and 5 to 6.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A000720 (pi).

Cf. A211005 (run lengths).

Same parity: A010051, A061007, A035026, A071574.

Programs

Haskell

a069754 1 = 0
a069754 2 = 1
a069754 n = 2 * a000720 n - 2 - (toInteger $ a010051 $ toInteger n)
-- Reinhard Zumkeller, Dec 04 2012

Mathematica

For[lst={0}; trans=0; n=2, n<100, n++, If[PrimeQ[n]!=PrimeQ[n-1], trans++ ]; AppendTo[lst, trans]]; lst
(* Second program: *)
pts[n_]:=Module[{c=2PrimePi[n]},If[PrimeQ[n],c-3,c-2]]; Join[{0,1},Array[ pts,80,3]] (* Harvey P. Dale, Nov 12 2011 *)
Accumulate[If[Sort[PrimeQ[#]]=={False,True},1,0]&/@Partition[ Range[ 0,80],2,1]] (* Harvey P. Dale, May 06 2013 *)

Formula

When n is prime, a(n) = 2*pi(n) - 3. When n is composite, a(n) = 2*pi(n) - 2. pi(n) is the prime counting function A000720.

For n > 2: a(n) = 2*A000720(n) - 2 - A010051(n). - Reinhard Zumkeller, Dec 04 2012