A095062 -id:A095062 - cp's OEIS frontend

A095006 Number of evil primes (A027699) in range ]2^n,2^(n+1)].

1, 1, 0, 3, 2, 5, 4, 23, 27, 62, 95, 222, 367, 777, 1269, 2910, 4859, 10140, 17714, 36714, 66020, 133400, 245959, 493532, 916913, 1822087, 3428633, 6782008, 12870735, 25339113, 48419194, 95194890, 182818705, 358637144, 691891351, 1355985684, 2625053871, 5142673207

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

			From _Michael De Vlieger_, Feb 27 2017: (Start)
a(2) = 1 since between 2^2 and 2^3 only the prime 5 (binary 11) has an even number of 1s.
a(3) = 0 since none of the primes between 2^3 and 2^4 have an even number of 1s in their binary expansions.
a(4) = 3 since the primes 17, 23, and 29 have an even number of 1s in their binary expansions (i.e., 10001, 10111, 11101). (End)

Antti Karttunen and John 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)].

Cf. A027699, A036378, A095005.

Table[m = Count[Prime@ Range[PrimePi[2^n] + 1, PrimePi[2^(n + 1) - 1]], k_ /; EvenQ@ DigitCount[k, 2, 1]]; Print@ m; m, {n, 24}] (* Michael De Vlieger, Feb 27 2017 *)

a(n) = A036378(n) - A095005(n).

a(34)-a(38) from Amiram Eldar, Jun 20 2024

A095056 Number of primes with three 1-bits (A081091) in range [2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 3, 0, 4, 2, 3, 2, 2, 2, 4, 1, 3, 4, 5, 3, 2, 1, 5, 1, 0, 2, 5, 2, 2, 8, 6, 0, 5, 3, 4, 2, 3, 2, 2, 0, 3, 5, 0, 1, 5, 3, 7, 0, 1, 2, 5, 1, 5, 2, 6, 0, 6, 0, 2, 3, 2, 1, 2, 0, 2, 3, 5, 3, 6, 2, 2, 2, 5, 2, 7, 1, 3, 2, 3, 1, 6, 2, 4, 3, 3, 2, 6, 1, 1, 5, 7, 2, 4, 2, 5, 0, 3, 4, 3, 1, 2, 1, 3, 0, 5

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

			From _Michael De Vlieger_, Feb 27 2017: (Start)
a(1) = 0 because there are no primes with three 1s in binary expansion between 2^1 and 2^2.
a(2) = 1 since the only prime between 2^2 and 2^3 with three 1s in binary expansion is 7 = binary 111.
a(3) = 2 since between 2^3 and 2^4 we have 11 and 13 (binary 1011 and 1101, respectively) have three 1s.
(End)

T. D. Noe, Table of n, a(n) for n=1..1000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence
Samuel S. Wagstaff, Jr., Prime Numbers with a fixed number of one bits or zero bits in their binary representation, Exp. Math. vol. 10, issue 2 (2001) 267, Table 2. - From _N. J. A. Sloane_, Jun 19 2011.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A081504, A095018, A095057.

Table[m = Count[Prime@ Range[PrimePi[2^n] + 1, PrimePi[2^(n + 1) - 1]], k_ /; DigitCount[k, 2, 1] == 3]; Print@ m; m, {n, 24}] (* Michael De Vlieger, Feb 27 2017 *)

More terms from T. D. Noe, Oct 17 2007

A095058 Number of primes with a single 0-bit (A095078) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 2, 2, 3, 0, 4, 4, 3, 1, 5, 1, 4, 0, 3, 2, 8, 1, 11, 4, 5, 0, 7, 1, 2, 0, 1, 5, 4, 0, 7, 5, 1, 1, 9, 0, 6, 0, 7, 1, 6, 0, 4, 7, 2, 1, 10, 3, 3, 1, 2, 1, 6, 0, 4, 3, 0, 1, 8, 3, 3, 0, 3, 1, 8, 1, 2, 2, 3, 0, 9, 1, 5, 2, 5, 8, 3, 0, 10, 3, 0, 2, 4, 4, 6, 1

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

For large n, the average value of a(n) is about 4. See A138290 for the n such that a(n)=0. - T. D. Noe, Mar 14 2008

T. D. Noe, Table of n, a(n) for n=1..2048
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence
Samuel S. Wagstaff, Jr., Prime Numbers with a fixed number of one bits or zero bits in their binary representation, Exp. Math. vol. 10, issue 2 (2001) 267, Table 3. - From _N. J. A. Sloane_, Jun 19 2011.
Samuel S. Wagstaff, Jr., Prime Numbers with a fixed number of one bits or zero bits in their binary representation, Exp. Math. vol. 10, issue 2 (2001) 267, Table 3.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A095018.

a(n) = sum(k=2^n+1, 2^(n+1), isprime(k) && (#select(x->x==0, binary(k))==1)); \\ Michel Marcus, Sep 11 2015

A095060 Number of fibeven primes (A095080) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 2, 2, 3, 3, 9, 16, 25, 50, 83, 150, 286, 540, 975, 1865, 3515, 6588, 12620, 23835, 45486, 86811, 165822, 317770, 608517, 1170182, 2254124, 4342530, 8383468, 16197159, 31335332, 60680818, 117633364, 228260489, 443281943, 861677274

Offset: 1

Antti Karttunen, Jun 01 2004

Antti Karttunen and John 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)].

Cf. A036378, A095061, A095062, A095067, A095080.

a(n) = A036378(n) - A095061(n) = A095062(n) + A095067(n).

a(34)-a(35) from Amiram Eldar, Jun 13 2024

A095061 Number of fibodd primes (A095081) in range [2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 2, 4, 4, 7, 18, 25, 54, 105, 178, 332, 637, 1165, 2194, 4161, 7770, 14800, 28100, 53525, 102394, 195938, 377301, 723938, 1391620, 2684760, 5178439, 10010119, 19362205, 37501838, 72702221, 141062816, 273985225, 532514962

Offset: 1

Antti Karttunen, Jun 01 2004

Antti Karttunen and John 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)].

Cf. A036378, A095060, A095066, A095069, A095081.

a(n) = A036378(n) - A095060(n) = A095066(n) + A095069(n).

a(34)-a(35) from Amiram Eldar, Jun 13 2024

A095073 Primes in whose binary expansion the number of 1-bits is one more than the number of 0-bits.

Original entry on oeis.org

5, 19, 71, 83, 89, 101, 113, 271, 283, 307, 313, 331, 397, 409, 419, 421, 433, 457, 1103, 1117, 1181, 1223, 1229, 1237, 1303, 1307, 1319, 1381, 1427, 1429, 1433, 1481, 1489, 1559, 1579, 1607, 1613, 1619, 1621, 1637, 1699, 1733, 1811, 1861

Offset: 1

Antti Karttunen, Jun 01 2004

			71 is in the sequence because 71_10 = 1000111_2. '1000111' has four 1's and three 0's. - _Indranil Ghosh_, Feb 03 2017

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

Intersection of A000040 and A031448. Subset of A095070. Cf. A095053.

Select[Prime[Range[500]], Differences[DigitCount[#, 2]] == {-1} &]

{ forprime(p=2, 2000,
  v=binary(p); s=0;
  for(k=1,#v, s+=if(v[k]==1,+1,-1));
  if(s==1,print1(p,", "))
) }

from sympy import isprime
i=1
j=1
while j<=25000:
    if isprime(i) and bin(i)[2:].count("1")-bin(i)[2:].count("0")==1:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017

A095074 Primes in whose binary expansion the number of 0-bits is less than or equal to number of 1-bits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 71, 79, 83, 89, 101, 103, 107, 109, 113, 127, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 283, 307, 311, 313, 317, 331, 347, 349, 359

Offset: 1

Antti Karttunen, Jun 01 2004

			From _Indranil Ghosh_, Feb 03 2017: (Start)
29 is in the sequence because 29_10 = 11101_2. '11101' has one 0 and three 1's.
37 is in the sequence because 37_10 = 100101_2. '100101' has three 1's and 3 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 A095071 in A000040. Differs from A057447 first time at n=18, where a(n)=71, while A057447(18)=67. Cf. A095054.

Select[Prime[Range[50]], DigitCount[#, 2, 0] <= DigitCount[#, 2, 1] &] (* Alonso del Arte, Jan 11 2011 *)

forprime(p=2,359,v=binary(p);s=0;for(k=1,#v,s+=if(v[k]==0,+1,-1));if(s<=0,print1(p,", "))) \\ Washington Bomfim, Jan 13 2011

from sympy import isprime
i=1
j=1
while j<=25000:
    if isprime(i) and bin(i)[2:].count("0")<=bin(i)[2:].count("1"):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017

A095077 Primes with four 1-bits in their binary expansion.

Original entry on oeis.org

23, 29, 43, 53, 71, 83, 89, 101, 113, 139, 149, 163, 197, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 523, 547, 593, 643, 673, 773, 1031, 1049, 1061, 1091, 1093, 1097, 1217, 1283, 1289, 1297, 1409, 1553, 1601, 2069, 2083, 2089, 2129

Offset: 1

Antti Karttunen, Jun 01 2004

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Subsequence of A027699. First differs from A085448 at n = 19, where a(n)=337, while A085448 continues from there with 311, whose binary expansion has six 1-bits, not four. Cf. A095057.
Cf. A000215 (primes having two bits set), A081091 (three bits set).
Cf. A264908.

Select[Prime[Range[320]], Plus@@IntegerDigits[#, 2] == 4 &] (* Alonso del Arte, Jan 11 2011 *)
Select[ Flatten[ Table[2^i + 2^j + 2^k + 1, {i, 3, 11}, {j, 2, i - 1}, {k, j - 1}]], PrimeQ] (* Robert G. Wilson v, Jul 30 2016 *)

bits1_4(x) = { nB = floor(log(x)/log(2)); z = 0;
for(i=0,nB,if(bittest(x,i),z++;if(z>4,return(0););););
if(z == 4, return(1);, return(0););};
forprime(x=17,2129,if(bits1_4(x),print1(x, ", ");););
\\ Washington Bomfim, Jan 11 2011

is(n)=isprime(n) && hammingweight(n)==4 \\ Charles R Greathouse IV, Jul 30 2016

list(lim)=my(v=List(),t); for(a=3,logint(lim\=1,2), for(b=2,a-1, for(c=1,b-1, t=1<lim, return(Vec(v))); if(isprime(t), listput(v,t))))); Vec(v) \\ Charles R Greathouse IV, Jul 30 2016

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A095077_gen(): # generator of terms
    return filter(isprime,map(lambda s:int('1'+''.join(s)+'1',2),(s for l in count(2) for s in multiset_permutations('0'*(l-2)+'11'))))
A095077_list = list(islice(A095077_gen(),30)) # Chai Wah Wu, Jul 19 2022

A095078 Primes with a single 0 bit in their binary expansion.

Original entry on oeis.org

2, 5, 11, 13, 23, 29, 47, 59, 61, 191, 223, 239, 251, 383, 479, 503, 509, 991, 1019, 1021, 2039, 3583, 3967, 4079, 4091, 4093, 6143, 15359, 16127, 16319, 16381, 63487, 65407, 65519, 129023, 131063, 245759, 253951, 261631, 261887, 262079

Offset: 1

Antti Karttunen, Jun 01 2004

Except for the first value 2, the sequence gives the primes of the form 2^k -2^j -1 with 0 < j < k-1. If j=k-1 we obtain the Mersenne primes. - Pierre CAMI, May 19 2005

{2} UNION (A000040 INTERSECT A190620). - Chai Wah Wu, Dec 19 2024

T. D. Noe, Table of n, a(n) for n=1..1000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A030130. Cf. A095058, A190620.

Select[Prime[Range[23000]],DigitCount[#,2,0]==1&] (* Harvey P. Dale, Nov 28 2019 *)

forprime(p=2,262079,v=binary(p);s=0;for(k=1,#v,s+=v[k]);if(#v-s==1,print1(p,", "))) \\ Washington Bomfim, Jan 13 2011

A095084 Fibevil primes, i.e., primes p whose Zeckendorf-expansion A014417(p) contains an even number of 1-fibits.

Original entry on oeis.org

7, 11, 23, 29, 37, 47, 53, 67, 83, 97, 101, 109, 127, 137, 139, 149, 157, 163, 193, 199, 223, 241, 263, 271, 277, 281, 283, 311, 317, 331, 337, 359, 373, 379, 389, 397, 409, 421, 439, 443, 461, 499, 503, 521, 547, 557, 563, 577, 593, 601, 607

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.

Intersection of A000040 and A095096.

Cf. A014417, A095083, A095064.

Select[Flatten[Position[Mod[DigitCount[Select[Range[0, 6000], BitAnd[#, 2 #] == 0 &], 2, 1], 2], 0]] - 1, PrimeQ] (* Amiram Eldar, Feb 07 2023 *)

cp's OEIS Frontend

A095006 Number of evil primes (A027699) in range ]2^n,2^(n+1)].

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A095056 Number of primes with three 1-bits (A081091) in range [2^n,2^(n+1)].

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A095058 Number of primes with a single 0-bit (A095078) in range ]2^n,2^(n+1)].

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A095060 Number of fibeven primes (A095080) in range ]2^n,2^(n+1)].

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A095061 Number of fibodd primes (A095081) in range [2^n,2^(n+1)].

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A095073 Primes in whose binary expansion the number of 1-bits is one more than the number of 0-bits.

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A095074 Primes in whose binary expansion the number of 0-bits is less than or equal to number of 1-bits.

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Python

A095077 Primes with four 1-bits in their binary expansion.

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