A100714 Number of runs in binary expansion of A000040(n) (the n-th prime number) for n > 0.

2, 1, 3, 1, 3, 3, 3, 3, 3, 3, 1, 5, 5, 5, 3, 5, 3, 3, 3, 3, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 1, 3, 5, 5, 7, 5, 5, 5, 5, 7, 5, 7, 3, 3, 5, 3, 5, 3, 3, 5, 5, 3, 3, 3, 3, 3, 5, 3, 7, 5, 5, 7, 5, 5, 5, 5, 7, 7, 7, 7, 5, 5, 5, 7, 5, 3, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 3, 5, 5, 3, 5, 3, 3, 5, 3, 3, 3, 5, 5, 5, 5, 7, 5, 5, 5

Offset: 1

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 11 2004

Keywords

• base
• easy
• nonn

Comments

Record values of a(n) = 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ... are set at the indices n = 1, 3, 12, 35, 121, 355, 1317, 4551, 15897, 56475, 197249, 737926, ... - R. J. Mathar, Mar 02 2007

Examples

			a(5)=3 because A000040(5) = 11_10 = 1011_2, which splits into three runs ({1}, {0}, {1,1}).
		

Links

• Eric Weisstein's World of Mathematics, Run-Length Encoding.

Crossrefs

Cf. A005811, A000040, A082554, A100722, A100723.

Programs

Maple

A100714 := proc(n)
    A005811(ithprime(n)) ;
end proc:
seq( A100714(n),n=1..105) ; # R. J. Mathar, Jul 08 2025

Mathematica

Table[Length[Split[IntegerDigits[Prime[n], 2]]], {n, 1, 128}]

PARI

a(n,p=prime(n))=hammingweight(bitxor(p, p>>1)) \\ Charles R Greathouse IV, Oct 19 2015

Python

from sympy import prime
def a(n): return ((p:=prime(n))^(p>>1)).bit_count()
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Feb 25 2023

Formula

a(n) = A005811(A000040(n)).