A095018 -id:A095018 - cp's OEIS frontend

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

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

A095019 Number of zero-bit dominant primes (A095071) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 3, 11, 10, 40, 52, 130, 154, 482, 649, 1756, 2483, 6479, 9640, 24022, 34812, 89306, 136739, 335115, 510833, 1265350, 1982321, 4781514, 7508064, 18079040, 28833595, 68709969, 110272081, 262002130, 425542739, 1000343760, 1632745091, 3828253857, 6305334325, 14683465908

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

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. A095018, A095071.

a(34)-a(40) from Amiram Eldar, Jun 13 2024

A095020 Number of one-bit dominant primes (A095070) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 2, 2, 4, 5, 10, 16, 32, 48, 97, 175, 334, 529, 1130, 1850, 3953, 6276, 13911, 23248, 49564, 81622, 178910, 300311, 650703, 1091809, 2380394, 4062176, 8780393, 15021634, 32618497, 56134342, 121625616, 209889612, 455265038, 791458830, 1711760073, 2982211935, 6457387921, 11302458576, 24430016732

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

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. A095018, A095070.

a(34)-a(40) from Amiram Eldar, Jun 13 2024

A095052 Number of primes with number of 0-bits equal to one plus number of 1-bits (A095072) in range ]2^2n,2^(2n+1)].

Original entry on oeis.org

0, 1, 3, 10, 25, 78, 283, 906, 3044, 10920, 37920, 135182, 487555, 1764216, 6415902, 23585285, 86789112, 320972293, 1192327462, 4441973622

Offset: 1

Antti Karttunen, Jun 01 2004

			In the range ]2^4,2^5] 17 (10001 in binary) is the only such prime thus a(2) = 1.

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. A095018, A095072.

a(17)-a(20) from Amiram Eldar, Jun 13 2024

A095053 Number of primes with number of 1-bits equal to one plus number 0-bits (A095073) in range ]2^2n,2^(2n+1)].

Original entry on oeis.org

1, 1, 5, 11, 28, 105, 362, 1093, 3659, 13001, 45171, 159510, 563833, 2008295, 7333827, 26730538, 97256891, 358079458, 1324674524, 4902380577

Offset: 1

Antti Karttunen, Jun 01 2004

			In the range ]2^2,2^3] 5 (101 in binary) is the only such prime thus a(1) = 1.
Similarly, in the range ]2^4,2^5] 19 (10011 in binary) is also unique in that respect, thus a(2) = 1 as well.

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. A095018, A095073.

a(17)-a(20) from Amiram Eldar, Jun 13 2024

A095057 Number of primes with four 1-bits (A095077) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 2, 2, 5, 4, 10, 6, 13, 11, 9, 16, 16, 18, 25, 15, 19, 15, 37, 17, 37, 29, 29, 32, 40, 23, 49, 31, 51, 39, 37, 30, 52, 46, 40, 42, 62, 43, 57, 42, 68, 52, 78, 60, 89, 54, 63, 59, 92, 58, 79, 82, 99, 73, 87, 47, 99, 74, 72, 81, 106, 56, 102, 85, 117, 85, 97, 64, 132, 93

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

T. D. Noe, Table of n, a(n) for n=1..200
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.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A095018, A095056, A095077.

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

A095054 Number of primes with number of 0-bits <= number of 1-bits (A095074) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 2, 2, 4, 7, 10, 20, 32, 65, 97, 203, 334, 718, 1130, 2381, 3953, 8266, 13911, 28995, 49564, 105524, 178910, 376969, 650703, 1383287, 2380394, 5044969, 8780393, 18699214, 32618497, 69349061, 121625616, 259051224, 455265038, 968649497, 1711760073, 3647018733, 6457387921, 13745846521, 24430016732

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

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. A095018, A095020, A095074.

a(n) = A095020(n) + (if n is odd) A095018((n+1)/2).

a(34)-a(40) from Amiram Eldar, Jun 13 2024

A095055 Number of primes with number of 1-bits <= number of 0-bits (A095075) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 7, 11, 27, 40, 80, 130, 343, 482, 1180, 1756, 4473, 6479, 15387, 24022, 58714, 89306, 213397, 335115, 802311, 1265350, 2965114, 4781514, 11185644, 18079040, 42048314, 68709969, 159433693, 262002130, 602733406, 1000343760, 2297551889, 3828253857, 8748722270, 14683465908

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

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. A095018, A095019, A095075.

a(n) = A095019(n) + (if n is odd) A095018((n+1)/2).

a(34)-a(40) from Amiram Eldar, Jun 13 2024

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

Original entry on oeis.org

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

cp's OEIS Frontend

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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PARI

A095019 Number of zero-bit dominant primes (A095071) in range ]2^n,2^(n+1)].

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A095020 Number of one-bit dominant primes (A095070) in range ]2^n,2^(n+1)].

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A095052 Number of primes with number of 0-bits equal to one plus number of 1-bits (A095072) in range ]2^2n,2^(2n+1)].

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A095053 Number of primes with number of 1-bits equal to one plus number 0-bits (A095073) in range ]2^2n,2^(2n+1)].

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A095057 Number of primes with four 1-bits (A095077) in range ]2^n,2^(n+1)].

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A095054 Number of primes with number of 0-bits <= number of 1-bits (A095074) in range ]2^n,2^(n+1)].

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A095055 Number of primes with number of 1-bits <= number of 0-bits (A095075) in range ]2^n,2^(n+1)].

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Keywords

Links

Crossrefs

Formula

Extensions

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

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Programs