A100714 -id:A100714 - cp's OEIS frontend

A082554 Primes whose base-2 representation is a block of 1's, followed by a block of 0's, followed by a block of 1's.

5, 11, 13, 17, 19, 23, 29, 47, 59, 61, 67, 71, 79, 97, 103, 113, 131, 191, 193, 199, 223, 227, 239, 241, 251, 257, 263, 271, 383, 449, 463, 479, 487, 499, 503, 509, 769, 911, 967, 991, 1009, 1019, 1021, 1031, 1039, 1087, 1151, 1279, 1543, 1567, 1663, 1823

Offset: 1

Randy L. Ekl, May 03 2003

The n-th prime is a term iff A100714(n) = 3. - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 11 2004

A019434 \{3} is a subsequence, since the base-2 representation of a Fermat prime 2^(2^k)+1 > 3 is a single 1, followed by a block of 2^k-1 0's, followed by a last single 1. - Bernard Schott, Mar 07 2023

			1987 = 11111000011_2, which is a block of 5 1's, followed by a block of 4 0's, followed by a block of 2 1's, so 1987 is a term.
a(3)=17 is a term because it is the 3rd prime whose binary representation splits into exactly three runs: 17_10 = 10001_2 splits into {{1}, {0,0,0}, {1}}.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Run-Length Encoding.

Cf. A100714, A000040. Primes in A043570.

Cf. A019434.

Select[Table[Prime[k], {k, 1, 500}], Length[Split[IntegerDigits[ #, 2]]] == 3 &]

decomp(s)=if(s%2==0,return(1),); k=1; while(k==1,k=s%2; s=floor(s/2)); if(s==0,return(1),); while(k==0,k=s%2; s=floor(s/2)); while(k==1,k=s%2; s=floor(s/2)); return(s)
forprime(i=1,2000,if(decomp(i)==0,print1(i,", ")))

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

A100723 Prime numbers whose binary representations are split into exactly seven runs.

Original entry on oeis.org

149, 173, 181, 277, 293, 331, 337, 347, 349, 373, 421, 557, 587, 593, 599, 601, 613, 617, 619, 653, 659, 673, 691, 701, 709, 727, 733, 757, 809, 811, 821, 857, 859, 877, 937, 941, 1061, 1069, 1093, 1097, 1117, 1129, 1163, 1171, 1181, 1187, 1201, 1213

Offset: 1

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

The n-th prime is a member iff A100714(n)=7

			a(3) = 181 is a member because it is the 3rd prime whose binary representation splits into exactly 7 runs: 181_10 = 10110101_2.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Run-Length Encoding.

Cf. A100714, A000040.

qprime:= proc(n) if isprime(n) then n fi end proc:
[seq(seq(seq(seq(seq(seq(seq(qprime(2^i1 - 2^i2 + 2^i3 - 2^i4 + 2^i5
- 2^i6 + 2^i7-1), i7 = 1..i6-1),i6=i5-1..2,-1),i5=3..i4-1),  i4=i3-1..4,-1),i3=5..i2-1),i2=i1-1..6,-1),i1=7..12)]; # Robert Israel, Nov 24 2020

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

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 &]

A100724 Prime numbers whose binary representations are split into at most 3 runs.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 47, 59, 61, 67, 71, 79, 97, 103, 113, 127, 131, 191, 193, 199, 223, 227, 239, 241, 251, 257, 263, 271, 383, 449, 463, 479, 487, 499, 503, 509, 769, 911, 967, 991, 1009, 1019, 1021, 1031, 1039, 1087, 1151, 1279, 1543, 1567

Offset: 1

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

The n-th prime is a term iff A100714(n) <= 3.

			a(3)=5 is a term because it is the 3rd prime whose binary representation splits into no more than 3 runs: 5_10 = 101_2.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Run-Length Encoding.

Includes A000668 and A095078.

Cf. A100714, A000040.

R:= 2,3: count:= 2:
for d from 2 while count < 100 do
  for a from d-1 to 1 by -1 do
    for b from 0 to a-1 do
      p:= 2*(2^d - 2^a + 2^b)-1;
      if isprime(p) then R:= R,p; count:= count+1 fi
  od od;
  p:= 2^(d+1)-1;
  if isprime(p) then R:= R,p; count:= count+1 fi
od:
R; # Robert Israel, Oct 30 2024

Select[Table[Prime[k], {k, 1, 50000}], Length[Split[IntegerDigits[ #, 2]]] <= 3 &]

A100725 Prime numbers whose binary representations are split into a maximum of 5 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, 151, 157, 163, 167, 179, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 307, 311

Offset: 1

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

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

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

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

Cf. A100714, A100722 (exactly 5 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

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A082554 Primes whose base-2 representation is a block of 1's, followed by a block of 0's, followed by a block of 1's.

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A100723 Prime numbers whose binary representations are split into exactly seven runs.

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

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A100724 Prime numbers whose binary representations are split into at most 3 runs.

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A100725 Prime numbers whose binary representations are split into a maximum of 5 runs.

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

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