A095077 -id:A095077 - cp's OEIS frontend

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

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

A081091 Primes of the form 2^i + 2^j + 1, i > j > 0.

Original entry on oeis.org

7, 11, 13, 19, 37, 41, 67, 73, 97, 131, 137, 193, 521, 577, 641, 769, 1033, 1153, 2053, 2081, 2113, 4099, 4129, 8209, 12289, 16417, 18433, 32771, 32801, 32833, 40961, 65539, 133121, 147457, 163841, 262147, 262153, 262657, 270337, 524353, 524801

Offset: 1

Reinhard Zumkeller, Mar 05 2003

This is sequence A070739 without the Fermat primes, A000215. Sequence A081504 lists the i for which there are no primes. - T. D. Noe, Jun 22 2007

Primes in A014311. - Reinhard Zumkeller, May 03 2012

			    7 = 2^2 + 2^1 + 1
   11 = 2^3 + 2^1 + 1
   13 = 2^3 + 2^2 + 1
   19 = 2^4 + 2^1 + 1
   37 = 2^5 + 2^2 + 1
   41 = 2^5 + 2^3 + 1
   67 = 2^6 + 2^1 + 1
   73 = 2^6 + 2^3 + 1
   97 = 2^6 + 2^5 + 1
  131 = 2^7 + 2^1 + 1
  137 = 2^7 + 2^3 + 1
  193 = 2^7 + 2^6 + 1
  521 = 2^9 + 2^3 + 1

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..7800 (n = 1..1000 from T. D. Noe)
Richard Ehrenborg and N. Bradley Fox, The Descent Set Polynomial Revisited, arXiv:1408.6858 [math.CO], 2014.
Norman B. Fox, Combinatorial Potpourri: Permutations, Products, Posets, and Pfaffians, University of Kentucky, Theses and Dissertations, Mathematics, Paper 25.

Essentially the same as A070739.
Cf. A000040, A000215, A081092, A010051.
Cf. A095077 (primes with four bits set).
A057733 = 2^A057732 + 3 and A039687 = 3*2^A002253 + 1 are subsequences.

a081091 n = a081091_list !! (n-1)
a081091_list = filter ((== 1) . a010051') a014311_list
-- Reinhard Zumkeller, May 03 2012

N:= 20: # to get all terms < 2^N
select(isprime, [seq(seq(2^i+2^j+1,j=1..i-1),i=1..N-1)]); # Robert Israel, May 17 2016

Select[Flatten[Table[2^i + 2^j + 1, {i, 21}, {j, i-1}]], PrimeQ] (* Alonso del Arte, Jan 11 2011 *)

do(mx)=my(v=List(),t); for(i=2,mx,for(j=1,i-1,if(ispseudoprime(t=2^i+2^j+1), listput(v,t)))); Vec(v) \\ Charles R Greathouse IV, Jan 02 2014

is(n)=hammingweight(n)==3 && isprime(n) \\ Charles R Greathouse IV, Aug 28 2017

A81091=[7]; next_A081091(p, i=exponent(p), j=exponent(p-2^i))=!until(isprime(2^i+2^j+1), j++>=i && i++ && j=1)+2^i+2^j
A081091(n)={for(k=#A81091, n-1, A81091=concat(A81091, next_A081091(A81091[k]))); A81091[n]} \\ M. F. Hasler, Mar 03 2023

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A081091_gen(): # generator of terms
    return filter(isprime,map(lambda s:int('1'+''.join(s)+'1',2),(s for l in count(1) for s in multiset_permutations('0'*(l-1)+'1'))))
A081091_list = list(islice(A081091_gen(),30)) # Chai Wah Wu, Jul 19 2022

A000120(a(n)) = 3.

A255564 Primes having in binary representation a nonprime number of 1's.

Original entry on oeis.org

2, 23, 29, 43, 53, 71, 83, 89, 101, 113, 139, 149, 163, 197, 263, 269, 277, 281, 293, 311, 317, 337, 347, 349, 353, 359, 373, 383, 389, 401, 449, 461, 467, 479, 503, 509, 523, 547, 571, 593, 599, 619, 643, 673, 683, 691, 739, 751, 773, 797, 811, 821, 839, 853, 857, 863, 881, 887, 907, 937, 977, 983, 991, 1013, 1019, 1021, 1031, 1049, 1061

Offset: 1

Antti Karttunen, May 14 2015

nonn

Equally: 2 followed by all primes with their hamming weight a composite number.

			2, which in binary (A007088) is "10", has just one 1-bit, and 1 is not a prime, thus 2 is included in the sequence.
23, which in binary is "10111", has four 1-bits, and 4 is not a prime, thus 23 is included in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement among primes: A081092.
Intersection of A000040 and A084345.
Subsequences: A027699 \ A019434, A085448, A095077, A255569.
Cf. A000120.

i = 0; forprime(n=2, 2^31, if(!isprime(hammingweight(n)), i++; write("b255564.txt", i, " ", n); if(i>=10000,return(n))));

;; With Antti Karttunen's IntSeq-library
(define A255564 (MATCHING-POS 1 1 (lambda (n) (and (prime? n) (not (prime? (A000120 n)))))))

A264908 Primes of the form 2^i + 2^j + 2^k - 1, i > j > k > 0.

Original entry on oeis.org

13, 37, 41, 43, 73, 83, 97, 103, 137, 139, 151, 163, 167, 193, 199, 223, 521, 523, 547, 577, 607, 641, 643, 647, 769, 1033, 1063, 1091, 1103, 1153, 1283, 1543, 1567, 1663, 2053, 2081, 2083, 2087, 2113, 2143, 2179, 2207, 2239, 2311, 2591, 2687, 3079, 3583, 4129, 4231, 4639

Offset: 1

Robert G. Wilson v, Nov 28 2015

nonn

Cf. A000668, A095077, A239712.

Select[ Flatten[ Table[2^i + 2^j + 2^k - 1, {i, 3, 10}, {j, 2, i - 1}, {k, j - 1}]], PrimeQ]

A362979 Square array, read by descending antidiagonals: row n lists the primes whose base-2 representation has exactly n ones, starting from n=3.

Original entry on oeis.org

7, 11, 23, 13, 29, 31, 19, 43, 47, 311

Offset: 3

Clark Kimberling, May 11 2023

			Corner:
  n=3:    7    11    13    19    37   41     67    73    97
  n=4:   23    29    43    53    71   83     89   101   113
  n=5:   31    47    59    61    79   103   107   109   151
  n=6:  311   317   347   349   359   373   461   467   571
The first four primes in row n=3 have these base-2 representations, respectively: 111, 1011, 1101, 10011.

Cf. A000040, A014499.

Cf. A019434 (row 2), A061712 (column 1), A081091 (row 3), A095077 (row 4).

t[n_] := Count[IntegerDigits[Prime[n], 2], 1]  (* A014499 *)
u = Table[t[n], {n, 1, 200}];
p[n_] := Flatten[Position[u, n]]
w = TableForm[Table[Prime[p[n]], {n, 3, 16}]]

New offset and edited by Michel Marcus, Jan 19 2024

cp's OEIS Frontend

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

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A081091 Primes of the form 2^i + 2^j + 1, i > j > 0.

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A255564 Primes having in binary representation a nonprime number of 1's.

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A264908 Primes of the form 2^i + 2^j + 2^k - 1, i > j > k > 0.

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Mathematica

A362979 Square array, read by descending antidiagonals: row n lists the primes whose base-2 representation has exactly n ones, starting from n=3.

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Mathematica

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