A082554 -id:A082554 - cp's OEIS frontend

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

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

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

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

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

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

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

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

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

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

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

A043570 Numbers whose base-2 representation has exactly 3 runs.

Original entry on oeis.org

5, 9, 11, 13, 17, 19, 23, 25, 27, 29, 33, 35, 39, 47, 49, 51, 55, 57, 59, 61, 65, 67, 71, 79, 95, 97, 99, 103, 111, 113, 115, 119, 121, 123, 125, 129, 131, 135, 143, 159, 191, 193, 195, 199, 207, 223, 225, 227, 231, 239, 241, 243, 247

Offset: 1

Clark Kimberling

Numbers of the form 2^n - 2^m + 2^k - 1 for n > m > k > 0. - Robert Israel, Jan 11 2018
A000051 \ {2, 3} is a subsequence, since the base-2 representation of a number of the form 2^k+1 > 3 consists of a single 1, followed by a block of k-1 0's, followed by a last single 1. Also, A000215 \ {3} is another subsequence, since the base-2 representation of a Fermat number 2^(2^k)+1 > 3 consists of a single 1, followed by a block of 2^k-1 0's, followed by a last single 1. - Bernard Schott, Mar 09 2023
Numbers k such that A005811(k) = 3. - Michel Marcus, Mar 10 2023

			115 = 1110011_2, which is a block of three 1's, followed by a block of two 0's, followed by a block of two 1's, so 115 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005811.
Cf. A000051, A000215.
Cf. A082554 (subsequence of primes).

seq(seq(seq(2^n-2^m+2^k-1, k=1..m-1),m=n-1..2,-1),n=2..10); # Robert Israel, Jan 11 2018

from itertools import count, islice
def agen(): yield from ((1<Michael S. Branicky, Feb 25 2023

A100722 Prime numbers whose binary representations are split into exactly five runs.

Original entry on oeis.org

37, 41, 43, 53, 73, 83, 89, 101, 107, 109, 137, 139, 151, 157, 163, 167, 179, 197, 211, 229, 233, 269, 281, 283, 307, 311, 313, 317, 353, 359, 367, 379, 389, 397, 401, 409, 419, 431, 433, 439, 443, 457, 461, 467, 491, 521, 523, 541, 547, 563, 569, 571, 577

Offset: 1

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

The n-th prime is a term iff A100714(n)=5.

			a(3)=43 is a term because it is the 3rd prime whose binary representation splits into exactly five runs. 43_10 = 101011_2 splits into {{1}, {0}, {1}, {0}, {1,1}}.

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

Cf. A100714, A000668 (exactly 1 run), A082554 (exactly 3 runs), A100723 (exactly 7 runs).

Select[Table[Prime[k], {k, 1, 50000}], Length[Split[IntegerDigits[ #, 2]]] == 5 &]

cp's OEIS Frontend

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

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A043570 Numbers whose base-2 representation has exactly 3 runs.

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A100722 Prime numbers whose binary representations are split into exactly five runs.

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Mathematica