A095079 -id:A095079 - cp's OEIS frontend

A095059 Number of primes with two 0-bits (A095079) in range ]2^n,2^(n+1)].

0, 0, 0, 1, 2, 4, 0, 9, 5, 14, 4, 16, 9, 18, 0, 21, 21, 21, 7, 41, 22, 31, 5, 37, 20, 33, 14, 37, 45, 47, 0, 69, 31, 36, 34, 55, 34, 71, 10, 60, 50, 69, 22, 81, 52, 59, 5, 97, 71, 79, 42, 67, 86, 95, 13, 103, 61, 81, 47, 98, 50, 110, 0, 108, 87, 116, 36, 125, 98, 98, 29, 126, 90, 125, 46, 107, 100, 125, 8, 158, 81, 109, 65, 156, 94, 131, 27, 127, 144, 146, 38, 167, 129, 137, 6, 127, 112, 178, 76

Offset: 1

Antti Karttunen, Jun 01 2004

Michael S. Branicky, 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 4. - 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. A095018, A095056, A095057, A095058.

from sympy import isprime
from itertools import combinations, count, islice
def a(n): # generator of terms
    if n < 2: return 0
    b, d = (1<Michael S. Branicky, Dec 27 2023

Added terms a(34)-a(99) from the Wagstaff paper. - N. J. A. Sloane, Jun 19 2011

A272478 Primes with a prime number of binary digits, and with a prime number of 1's and a prime number of 0's.

Original entry on oeis.org

17, 19, 79, 103, 107, 109, 5119, 6079, 6911, 7039, 7103, 7151, 7159, 7919, 7927, 7933, 8059, 8111, 8123, 8167, 8171, 8179, 442367, 458239, 458719, 458747, 487423, 491503, 499711, 507839, 507901, 515839, 516091, 520063, 523007, 523261, 523519, 523759, 523771, 523903, 524219, 524221, 524269

Offset: 1

Andres Cicuttin, May 01 2016

If the sum of primes p and q is a prime r, then one of p and q must be 2. - N. J. A. Sloane, May 01 2016

			a(3) = 79, its binary representation is 1001111 with (prime) 7 digits, (prime) 5 1's and (prime) 2 0's.

Alois P. Heinz, Table of n, a(n) for n = 1..10859
Michel Marcus, 3 primes

Cf. A052294, A095079, A272441.

Cf. A006512 (bitlengths of terms).

Select[Table[Prime[j],{j,1,120000}],PrimeQ[Total@IntegerDigits[#,2]]&&PrimeQ[Length@IntegerDigits[#,2]]&&PrimeQ[(Length@IntegerDigits[#,2]-Total@IntegerDigits[#,2])]&]
Select[Prime@ Range[10^5], And[PrimeQ@ Total@ #, PrimeQ@ First@ #, PrimeQ@ Last@ #] &@ DigitCount[#, 2] &] (* Michael De Vlieger, May 01 2016 *)

isok(n) = isprime(n) && isprime(#binary(n)) && isprime(hammingweight(n)) && isprime(#binary(n) - hammingweight(n)); \\ Michel Marcus, May 01 2016

from sympy import isprime, nextprime
from itertools import combinations, islice
def agen(): # generator of terms
    p = 2
    while True:
        p = nextprime(p)
        if not isprime(p+2): continue
        if isprime(t:=(1<<(p+1))+1): yield t
        b = (1<<(p+2))-1
        for i, j in combinations(range(p), 2):
            if isprime(t:=b-(1<<(p-i))-(1<<(p-j))):
                yield t
print(list(islice(agen(), 43))) # Michael S. Branicky, Dec 27 2023

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A095059 Number of primes with two 0-bits (A095079) in range ]2^n,2^(n+1)].

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A272478 Primes with a prime number of binary digits, and with a prime number of 1's and a prime number of 0's.

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