A022447 Fractal sequence of the dispersion of the primes.

1, 1, 1, 2, 1, 3, 2, 4, 5, 6, 1, 7, 3, 8, 9, 10, 2, 11, 4, 12, 13, 14, 5, 15, 16, 17, 18, 19, 6, 20, 1, 21, 22, 23, 24, 25, 7, 26, 27, 28, 3, 29, 8, 30, 31, 32, 9, 33, 34, 35, 36, 37, 10, 38, 39, 40, 41, 42, 2, 43, 11, 44, 45, 46, 47, 48, 4, 49, 50, 51, 12, 52, 13, 53, 54, 55, 56, 57, 14

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Examples

			From _Sean A. Irvine_, May 20 2019: (Start)
The prime counting function, pi(n), is iterated (possibly zero times) until a nonprime is reached.  If the result of this iteration is m, then a(n) = m - pi(m).  Examples:
n=11: pi(11)=5, pi(5)=3, pi(3)=2, pi(2)=1. Hence, m=1 and so a(11) = 1-pi(1) = 1.
n=12: is already nonprime, hence m=12 and so a(12) = 12-pi(12) = 7.
n=13: pi(13)=6 (a nonprime), hence m=6 and so a(13) = 6-pi(6) = 3.
(End)
		

References

• C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria, vol. 45, p. 157, 1997.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Sean A. Irvine, Java program (github)
• C. Kimberling, Interspersions

Crossrefs

Cf. A000720, A022446, A114537, A114538, A114577.

Programs

Mathematica

m = 30; list = Table[Length[NestWhileList[PrimePi, n, PrimeQ]], {n, m}]; Table[Length@Position[Take[list, k], list[[k]]], {k, Length[list]}] (* Birkas Gyorgy, Apr 04 2011 *)
primefractal[n_]:= (# - PrimePi[#]) &@NestWhile[PrimePi, n, PrimeQ]; Array[primefractal, 30] (* Birkas Gyorgy, Apr 04 2011 *)

Extensions

Terms a(67) onward added by G. C. Greubel, Feb 28 2018

Offset corrected by Sean A. Irvine, May 20 2019