A100723 -id:A100723 - 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)).

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 &]

A100726 Prime numbers whose binary representations are split into a maximum of 7 runs.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

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

The m-th prime is a term iff A100714(m) <= 7.

Missing primes begin 661, 677, 683, 853, 1109, 1193, 1237, 1301, 1321, 1361, 1367, 1373, .... - Charles R Greathouse IV, Oct 19 2015

			a(3)=5 is a term because it is the 3rd prime whose binary representation splits into at most 7 runs: 5_10 = 101_2.

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

Cf. A100714, A100725 (maximum of 5 runs), A100724 (maximum of 3 runs), A100723 (exactly 7 runs).

Select[Table[Prime[k], {k, 1, 50000}], Length[Split[IntegerDigits[ #, 2]]] <= 7 &]

is(n)=hammingweight(bitxor(n, n>>1))<8 && isprime(n) \\ Charles R Greathouse IV, Oct 19 2015

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

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A100726 Prime numbers whose binary representations are split into a maximum of 7 runs.

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Mathematica

PARI