A095315 -id:A095315 - cp's OEIS frontend

A095335 Number of A095315-primes in range ]2^n,2^(n+1)].

1, 1, 2, 2, 4, 8, 19, 22, 43, 68, 159, 235, 537, 844, 1914, 2849, 6378, 10138, 23145, 37023, 82295, 134477, 303833, 494625, 1098521, 1829183, 4075501, 6789809, 15062515, 25412867, 56401388, 95440507, 210680037

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 1, 0.5, 1, 0.4, 0.571429, 0.615385, 0.826087, 0.511628, 0.573333, 0.49635, 0.623529, 0.506466, 0.615826, 0.523573, 0.631683, 0.499037, 0.593358, 0.497205, 0.599068, 0.503126, 0.586414, 0.501376, 0.591451, 0.501741, 0.579964, 0.501731, 0.579953, 0.500653, 0.574745, 0.501264, 0.574454, 0.501433, 0.570449

Ratios a(n)/A095296(n) converge as: 1, 1, 1, 0.666667, 0.8,1.6, 1.1875, 1.047619, 0.895833, 0.985507, 0.908571, 1.026201,1.015123, 1.098958, 1.034595, 0.996154, 1.016252, 0.98888, 0.99557,1.012581, 1.008245, 1.005518, 1.011728, 1.006987, 1.006148, 1.006948,1.00328, 1.002615, 1.002721, 1.00507, 1.004757, 1.005748, 1.003766

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095334(n).

A095314 Primes in whose binary expansion the number of 1 bits is > 2 + number of 0 bits.

Original entry on oeis.org

7, 23, 29, 31, 47, 59, 61, 79, 103, 107, 109, 127, 191, 223, 239, 251, 311, 317, 347, 349, 359, 367, 373, 379, 383, 431, 439, 443, 461, 463, 467, 479, 487, 491, 499, 503, 509, 607, 631, 701, 719, 727, 733, 743, 751, 757, 761, 823, 827, 829, 859

Offset: 1

Antti Karttunen, Jun 04 2004

Robert Israel, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence

Complement of A095315 in A000040. Subset of A095286. Subset: A095318. Cf. also A095334.

f:= proc(n) local L,d,s;
    if not isprime(n) then return false fi;
    L:= convert(n,base,2);
    convert(L,`+`) > nops(L)/2+1
end proc:
select(f, [seq(i,i=3..1000,2)]); # Robert Israel, Oct 26 2023

n1Q[p_]:=Module[{be=IntegerDigits[p,2]},Total[be]>2+Count[be,0]]; Select[ Prime[ Range[150]],n1Q] (* Harvey P. Dale, Oct 26 2022 *)

B(x) = { nB = floor(log(x)/log(2)); b1 = 0; b0 = 0;
for(i = 0, nB, if(bittest(x,i), b1++;, b0++;); );
if(b1 > (2+b0), return(1);, return(0););};
forprime(x = 2, 859, if(B(x), print1(x, ", "); ); );
\\ Washington Bomfim, Jan 12 2011

A095287 Primes in whose binary expansion the number of 1-bits is <= 1 + number of 0-bits.

Original entry on oeis.org

2, 5, 17, 19, 37, 41, 67, 71, 73, 83, 89, 97, 101, 113, 131, 137, 139, 149, 163, 193, 197, 257, 263, 269, 271, 277, 281, 283, 293, 307, 313, 331, 337, 353, 389, 397, 401, 409, 419, 421, 433, 449, 457, 521, 523, 541, 547, 557, 563, 569, 577, 587, 593, 601, 613

Offset: 1

Antti Karttunen, Jun 04 2004

			From _Indranil Ghosh_, Feb 03 2017: (Start)
5 is in the sequence because 5_10 = 101_2. '101' has two 1's and one 0.
17 is in the sequence because 17_10 = 10001_2. '10001' has two 1's and three 0's. (End)

Indranil Ghosh, Table of n, a(n) for n = 1..25000
A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence

Complement of A095286 in A000040. Subset: A095075. Subset of A095315. Cf. also A095297.

Select[Prime[Range[200]],DigitCount[#,2,1]<=1+DigitCount[#,2,0]&] (* Harvey P. Dale, Apr 18 2023 *)

forprime(p=2,613,v=binary(p);s=0;for(k=1,#v,s+=if(v[k]==1,+1,-1));if(s<=1,print1(p,", "))) \\ Washington Bomfim, Jan 13 2011

from sympy import isprime
i=1
j=1
while j<=250:
    if isprime(i) and bin(i)[2:].count("1")<=1+bin(i)[2:].count("0"):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017

A095319 Primes in whose binary expansion the number of 1 bits is <= 3 + number of 0 bits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 53, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 193, 197, 199, 211, 227, 229, 233, 241, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311

Offset: 1

Antti Karttunen

Differs from primes (A000040) first time at n=11, where a(11)=37, while A000040(11)=31, as 31 whose binary expansion is 11111, with 5 1-bits and no 0 bits is the first prime excluded from this sequence. - Jun 04 2004

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence

Complement of A095318 in A000040. Subset of A095323, subset: A095315. A095329.

Select[Prime[Range[70]],DigitCount[#,2,1]Harvey P. Dale, Feb 22 2020 *)

B(x) = { nB = floor(log(x)/log(2)); b1 = 0; b0 = 0;
for(i = 0, nB, if(bittest(x,i), b1++;, b0++;); );
if(b1 <= (3+b0), return(1);, return(0););};
forprime(x = 2, 311, if(B(x), print1(x, ", "); ); );
\\ Washington Bomfim, Jan 12 2011

cp's OEIS Frontend

A095335 Number of A095315-primes in range ]2^n,2^(n+1)].

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A095314 Primes in whose binary expansion the number of 1 bits is > 2 + number of 0 bits.

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A095287 Primes in whose binary expansion the number of 1-bits is <= 1 + number of 0-bits.

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A095319 Primes in whose binary expansion the number of 1 bits is <= 3 + number of 0 bits.

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Author

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Programs

Mathematica

PARI