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

1,4

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

Links

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000720 (pi).

Cf. A211005 (run lengths).

Same parity: A010051, A061007, A035026, A071574.

Sequence in context: A102674 A097623 A198462 * A097622 A236561 A352499

Adjacent sequences: A069751 A069752 A069753 * A069755 A069756 A069757

Keyword

easy,nice,nonn

Author

T. D. Noe, May 02 2002

Status

approved