A081091 -id:A081091 - 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

A360448 Indices of primes of the form p = 2^i + 2^j + 1, i > j > 0 (A081091).

Original entry on oeis.org

4, 5, 6, 8, 12, 13, 19, 21, 25, 32, 33, 44, 98, 106, 116, 136, 174, 191, 310, 313, 319, 565, 568, 1029, 1470, 1902, 2111, 3513, 3518, 3521, 4289, 6544, 12426, 13632, 15000, 23001, 23003, 23043, 23673, 43395, 43420, 43465, 45859, 62947, 82029, 82063, 91466, 155612, 155900, 295957, 564164

Offset: 1

M. F. Hasler, Mar 03 2023

nonn

Cf. A000720 (prime counting function), A081091 (primes of the form 2^i + 2^j + 1, i > j > 0), A014499 (Hamming weight of the n-th prime), A000040 (the primes), A000120 (Hamming weight).

Position[Prime[Range[600000]], ?(DigitCount[#, 2, 1] == 3 &)] // Flatten (* _Amiram Eldar, Mar 04 2023 *)
PrimePi@ Union@ Select[Flatten@ Table[2^i + 2^j + 1, {i, 0, 23}, {j, 0, i - 1}], PrimeQ] (* Michael De Vlieger, Mar 21 2023 *)

A360448(n) = primepi(A081091(n))
is_A360448(n) = hammingweight(prime(n))==3

from itertools import count, islice
from sympy import primepi, isprime
def A360448_gen(): # generator of terms
    for i in count(2):
        k = (1<A360448_list = list(islice(A360448_gen(),20)) # Chai Wah Wu, Mar 21 2023

a(n) = A000720(A081091(n)).

This sequence = { n | A000120(A000040(n)) = 3 }.

A014499 Number of 1's in binary representation of n-th prime.

Original entry on oeis.org

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

Offset: 1

Ingemar Assarsjo (ingemar(AT)binomen.se)

a(n) is the rank of prime(n) in the base-2 dominance order on the natural numbers. - Tom Edgar, Mar 25 2014

			From _M. F. Hasler_, Mar 03 2023: (Start)
a(n) = 1 only for p(n = 1) = 2, the only prime equal to a power of 2.
a(n) = 2 for n in A159611 = A000720(A019434) = {2, 3, 7, 55, 6543} (probably complete), the Fermat primes F[k] = 2^2^k + 1 with k = 0, 1, 2, 3, 4. (On the graph one can distinctly see a(6543) = 2 corresponding to F[4] = 65537.)
a(n) = 3 for n in A000720(A081091) = (4, 5, 6, 8, 12, 13, 19, 21, 25, 32, 33, 44, 98, 106, 116, 136, 174, 191, 310, 313, 319, 565, 568, ...). (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.
Christian Elsholtz, Almost all primes have a multiple of small Hamming weight, arXiv:1602.05974 [math.NT], 2016.
Index entries for sequences related to binary expansion of n

Cf. A035103, A035100, A004676, A090455.
Cf. A027697, A027699
Cf. A180024. - Reinhard Zumkeller, Aug 08 2010
Cf. A072084.
Cf. A159611 (indices of 2s), A000720(A081091) (indices of 3s). - M. F. Hasler, Mar 03 2023

a014499 = a000120 . a000040  -- Reinhard Zumkeller, Feb 10 2013

[&+Intseq(NthPrime(n), 2): n in [1..100] ]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 2], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

A014499(n)=hammingweight(prime(n)) \\ M. F. Hasler, Nov 20 2009, updated Mar 03 2023

from sympy import prime
def A014499(n): return prime(n).bit_count() # Chai Wah Wu, Mar 22 2023

[sum(i.digits(base=2)) for i in primes_first_n(200)] # Tom Edgar, Mar 25 2014

a(n) = A000120(A000040(n)).
a(A049084(A061712(n))) = n. - Reinhard Zumkeller, Feb 10 2013
a(n) = [x^prime(n)] (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018

A014311 Numbers with exactly 3 ones in binary expansion.

Original entry on oeis.org

7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 131, 133, 134, 137, 138, 140, 145, 146, 148, 152, 161, 162, 164, 168, 176, 193, 194, 196, 200, 208, 224, 259, 261, 262, 265, 266, 268, 273, 274, 276, 280, 289, 290, 292, 296, 304

Offset: 1

Al Black (gblack(AT)nol.net)

Equivalently, sums of three distinct powers of 2.
Appears to give all n such that 64 is the highest power of 2 dividing A005148(n). - Benoit Cloitre, Jun 22 2002
From Gus Wiseman, Oct 05 2020: (Start)
These are numbers k such that the k-th composition in standard order has length 3. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. The sequence together with the corresponding standard compositions begins:
(1,1,1)     44: (2,1,3)     97: (1,5,1)
(2,1,1)     49: (1,4,1)     98: (1,4,2)
(1,2,1)     50: (1,3,2)    100: (1,3,3)
(1,1,2)     52: (1,2,3)    104: (1,2,4)
(3,1,1)     56: (1,1,4)    112: (1,1,5)
(2,2,1)     67: (5,1,1)    131: (6,1,1)
(2,1,2)     69: (4,2,1)    133: (5,2,1)
(1,3,1)     70: (4,1,2)    134: (5,1,2)
(1,2,2)     73: (3,3,1)    137: (4,3,1)
(1,1,3)     74: (3,2,2)    138: (4,2,2)
(4,1,1)     76: (3,1,3)    140: (4,1,3)
(3,2,1)     81: (2,4,1)    145: (3,4,1)
(3,1,2)     82: (2,3,2)    146: (3,3,2)
(2,3,1)     84: (2,2,3)    148: (3,2,3)
(2,2,2)     88: (2,1,4)    152: (3,1,4)
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.
Stephen Morley, HAKMEM Item 175 (Gosper).
Tilman Piesk, First 56 elements in a tetrahedral array.

Cf. A038465 (base 3), A038471 (base 4), A038475 (base 5).
Cf. A081091 (primes), A212190 (squares), A212192 (triangular numbers), A173589 (Fibbinary).
Cf. A057168.
Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hammingweight = 1, 2, ..., 9).
Cf. A056558, A194848, A194847.
A000217(n-2) counts compositions into three parts.
A001399(n-3) = A069905(n) = A211540(n+2) counts the unordered case.
A001399(n-6) = A069905(n-3) = A211540(n-1) counts the unordered strict case.
A001399(n-6)*6 = A069905(n-3)*6 = A211540(n-1)*6 counts the strict case.
A014612 is an unordered version, with strict case A007304.
A337453 is the strict case.
A337461 counts the coprime case.
A033992 lists numbers divisible by exactly three different primes.
A323024 lists numbers with exactly three different prime multiplicities.
Cf. A220377, A307534, A337459, A337460, A337561, A337603, A337604.

unsigned hakmem175(unsigned x) {
    unsigned s, o, r;
    s = x & -x;  r = x + s;
    o = r ^ x;  o = (o >> 2) / s;
    return r | o;
}
unsigned A014311(int n) {
    if (n == 1) return 7;
    return hakmem175(A014311(n - 1));
}  // Peter Luschny, Jan 01 2014

a014311 n = a014311_list !! (n-1)
a014311_list = [2^x + 2^y + 2^z |
                x <- [2..], y <- [1..x-1], z <- [0..y-1]]
-- Reinhard Zumkeller, May 03 2012

Select[Range[200], (Count[IntegerDigits[#, 2], 1] == 3)&]
nn = 8; Flatten[Table[2^i + 2^j + 2^k, {i, 2, nn}, {j, 1, i - 1}, {k, 0, j - 1}]] (* T. D. Noe, Nov 05 2013 *)

for(n=0,10^3,if(hammingweight(n)==3,print1(n,", "))); \\ Joerg Arndt, Mar 04 2014

print1(t=7);for(i=2,50,print1(","t=A057168(t))) \\ M. F. Hasler, Aug 27 2014

A014311_list = [2**a+2**b+2**c for a in range(2,6) for b in range(1,a) for c in range(b)] # Chai Wah Wu, Jan 24 2021

from itertools import islice
def A014311_gen(): # generator of terms
    yield (n:=7)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A014311_list = list(islice(A014311_gen(),20)) # Chai Wah Wu, Mar 10 2025

from math import isqrt, comb
from sympy import integer_nthroot
def A014311(n): return (1<<(r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+(1<<(a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1))+(1<Chai Wah Wu, Mar 10 2025

A000120(a(n)) = 3. - Reinhard Zumkeller, May 03 2012
Start with A084468. If n is in sequence, then 2n is too. - Ralf Stephan, Aug 16 2013
a(n+1) = A057168(a(n)). - M. F. Hasler, Aug 27 2014
a(n) = 2^A056558(n-1) + 2^A194848(n-1) + 2^A194847(n-1). - Ridouane Oudra, Sep 06 2020
Sum_{n>=1} 1/a(n) = A367110 = 1.428591545852638123996854844400537952781688750906133068397189529775365950039... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Extension and program by Olivier Gérard

A039687 Primes of the form 3*2^k + 1.

Original entry on oeis.org

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657, 221360928884514619393, 2353913150770005286438421033702874906038383291674012942337

Offset: 1

Felice Russo

nonn

Primes of the form 6m+1 (A002476) of A074781. - Bernard Schott, Dec 14 2020

S. W. Golomb, Properties of the sequence 3*2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3*2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

For more terms see A002253. These are the primes in A004119 (or A181565).
Cf. A002476, A074781.
Subsequence of A081091.

Select[Table[6 2^n+1,{n,0,250}],PrimeQ] (* Harvey P. Dale, Dec 17 2010 *)

A039687(n)=3<<A002253(n)+1 \\ M. F. Hasler, Mar 03 2023

a(n) = 3*2^A002253(n) + 1. - M. F. Hasler, Mar 03 2023

A081092 Primes having a prime number of 1's in their binary representation.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 47, 59, 61, 67, 73, 79, 97, 103, 107, 109, 127, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443

Offset: 1

Reinhard Zumkeller, Mar 05 2003

Same as primes with prime binary digit sum.
Primes with prime decimal digit sum are A046704.
Sum_{a(n) < x} 1/a(n) is asymptotic to log(log(log(x))) as x -> infinity; see Harman (2012). Thus the sequence is infinite. - Jonathan Sondow, Jun 09 2012
A049084(A000120(a(n))) > 0; A081091, A000215 and A081093 are subsequences.

			15th prime = 47 = '101111' with five 1's, therefore 47 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. Harman, Counting Primes whose Sum of Digits is Prime, J. Integer Seq., 15 (2012), Article 12.2.2.

Cf. A000040, A000120, A046704, A081093.

Subsequence of A052294.

a081092 n = a081092_list !! (n-1)
a081092_list = filter ((== 1) . a010051') a052294_list
-- Reinhard Zumkeller, Nov 16 2012

q:= n-> isprime(n) and isprime(add(i,i=Bits[Split](n))):
select(q, [$1..500])[];  # Alois P. Heinz, Sep 28 2023

Clear[BinSumOddQ];BinSumPrimeQ[a_]:=Module[{i,s=0},s=0;For[i=1,i<=Length[IntegerDigits[a,2]],s+=Extract[IntegerDigits[a,2],i];i++ ];PrimeQ[s]]; lst={};Do[p=Prime[n];If[BinSumPrimeQ[p],AppendTo[lst,p]],{n,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 06 2009 *)
Select[Prime[Range[100]], PrimeQ[Apply[Plus, IntegerDigits[#, 2]]] &] (* Jonathan Sondow, Jun 09 2012 *)

lista(nn) = {forprime(p=2, nn, if (isprime(hammingweight(p)), print1(p, ", ")););} \\ Michel Marcus, Jan 16 2015

from sympy import isprime
def ok(n): return isprime(n.bit_count()) and isprime(n)
print([k for k in range(444) if ok(k)]) # Michael S. Branicky, Dec 27 2023

A239712 Primes of the form m = 2^i + 2^j - 1, where i > j >= 0.

Original entry on oeis.org

2, 5, 11, 17, 19, 23, 47, 67, 71, 79, 131, 191, 257, 263, 271, 383, 1031, 1039, 1087, 1151, 1279, 2063, 2111, 4099, 4111, 4127, 4159, 5119, 6143, 8447, 16447, 20479, 32771, 32783, 32831, 33023, 33791, 65537, 65539, 65543, 65551, 65599, 66047, 73727, 81919, 262147, 262151, 262271, 262399, 263167

Offset: 1

Hieronymus Fischer, Mar 28 2014 and Apr 22 2014

nonn

Numbers m such that b = 2 is the only base such that the base-b digital sum of m + 1 is equal to b.
Example: 5 + 1 = 110_2 which implies ds_2(5 + 1) = 2 = b, where ds_b = digital sum in base-b. However, ds_3(6) = 2 <> 3, ds_4(6) = 3 <> 4, ds_5(6) = 2 <> 5, ds_6(6) = 1 <> 6. For all other bases > 6 we have ds_b(6) = 6 <> b. It follows that b = 2 is the only such base.
The base-2 representation of a term 2^i + 2^j - 1 has a base-2 digital sum of 1 + j.
In base-2 representation the first terms are 10, 101, 1011, 10001, 10011, 10111, 101111, 1000011, 1000111, 1001111, 10000011, 10111111, 100000001, 100000111, 100001111, 101111111, 10000000111, 10000001111, 10000111111, 10001111111, ...
Numbers m = 2^i + 2^j - 1 with odd i and j are not terms. Example: 10239 = 2^13 + 2^11 - 1 is not a prime.

			a(1) = 2, since 2 = 2^1 + 2^0 - 1 is prime.
a(5) = 19, since 19 = 2^4 + 2^2 - 1 is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..250

Cf. A007953, A018900, A081091, A008864, A187813.

Cf. A239703, A239708, A239709, A239713 - A239720.

Select[Union[Total/@(2^#&/@Subsets[Range[0,20],{2}])-1],PrimeQ] (* Harvey P. Dale, Aug 08 2014 *)

A239712
"Answers the n-th term of A239712.
  Usage: n A239712
  Answer: a(n)"
  | a b i k m p q terms |
  terms := OrderedCollection new.
  b := 2.
  p := 1.
  k := 0.
  m := 0.
  [k < self] whileTrue:
         [m := m + 1.
         p := b * p.
         q := 1.
         i := 0.
         [i < m and: [k < self]] whileTrue:
                   [i := i + 1.
                   a := p + q - 1.
                   a isPrime
                        ifTrue:
                            [k := k + 1.
                            terms add: a].
                   q := b * q]].
  ^terms at: self
[by Hieronymus Fischer, Apr 22 2014]
-----------

floorPrimesWhichAreDistinctPowersOf: b withOffset: d
  "Answers an array which holds the primes < n that obey b^i + b^j + d, i>j>=0,
  where n is the receiver. b > 1 (here: b = 2, d = -1).
  Uses floorDistinctPowersOf: from A018900
  Usage:
  n floorPrimesWhichAreDistinctPowersOf: b withOffset: d
  Answer: #(2 5 11 17 19 23 ...) [terms < n]"
  ^((self - d floorDistinctPowersOf: b)
  collect: [:i | i + d]) select: [:i | i isPrime]
[by Hieronymus Fischer, Apr 22 2014]
------------

primesWhichAreDistinctPowersOf: b withOffset: d
  "Answers an array which holds the n primes of the form b^i + b^j + d, i>j>=0, where n is the receiver.
  Direct calculation by scanning b^i + b^j + d in increasing order and selecting terms which are prime.
  b > 1; this sequence: b = 2, d = 1.
  Usage:
  n primesWhichAreDistinctPowersOf: b withOffset: d
  Answer: #(2 5 11 17 19 23 ...) [a(1) ... a(n)]"
  | a k p q terms n |
  terms := OrderedCollection new.
  n := self.
  k := 0.
  p := b.
  [k < n] whileTrue:
         [q := 1.
         [q < p and: [k < n]] whileTrue:
                   [a := p + q + d.
                   a isPrime
                        ifTrue:
                            [k := k + 1.
                            terms add: a].
                   q := b * q].
         p := b * p].
  ^terms asArray
[by Hieronymus Fischer, Apr 22 2014]

a(n) = A239708(n) - 1.

a(n+1) = min(A018900(k) > a(n)| A018900(k) - 1 is prime, k >= 1) - 1.

Examples moved from Maple field to Examples field by Harvey P. Dale, Aug 08 2014

A334092 Primes p of the form of the form q*2^h + 1, where q is one of the Fermat primes; Primes p for which A329697(p) == 2.

Original entry on oeis.org

7, 11, 13, 41, 97, 137, 193, 641, 769, 12289, 40961, 163841, 557057, 786433, 167772161, 2281701377, 3221225473, 206158430209, 2748779069441, 6597069766657, 38280596832649217, 180143985094819841, 221360928884514619393, 188894659314785808547841, 193428131138340667952988161

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p such that p-1 is not a power of two, but for which A171462(p-1) = (p-1-A052126(p-1)) is [a power of 2].

Primes of the form ((2^(2^k))+1)*2^h + 1, where ((2^(2^k))+1) is one of the Fermat primes, A019434, 3, 5, 17, 257, ..., .

Charles R Greathouse IV, Table of n, a(n) for n = 1..53

Primes in A334102.
Intersection of A081091 and A147545.
Subsequences: A039687, A050526, A300407.
Cf. A000040, A010051, A019434, A052126, A171462, A329697. Cf. also A334093, A334094, A334095, A334096.

isA334092(n) = (isprime(n)&&2==A329697(n));

A052126(n) = if(1==n,n,n/vecmax(factor(n)[, 1]));
A209229(n) = (n && !bitand(n,n-1));
isA334092(n) = (isprime(n)&&(!A209229(n-1))&&A209229(n-1-A052126(n-1)));

list(lim)=if(exponent(lim\=1)>=2^33, error("Verify composite character of more Fermat primes before checking this high")); my(v=List(),t); for(e=0,4, t=2^2^e+1; while((t<<=1)Charles R Greathouse IV, Apr 14 2020

More terms from Giovanni Resta, Apr 14 2020

A081504 Numbers n such that there are no primes of the form 2^n+2^i+1 for 0

Original entry on oeis.org

8, 25, 32, 40, 43, 48, 56, 58, 64, 96, 104, 112, 120, 128, 134, 140, 145, 152, 160, 176, 185, 192, 208, 212, 224, 235, 240, 244, 248, 252, 256, 264, 272, 280, 286, 288, 292, 302, 304, 308, 320, 326, 332, 348, 356, 360, 384, 392, 394, 400, 416, 418, 432, 448

Offset: 1

Ralf Stephan, Apr 21 2003

nonn

There seem to be no such numbers (bit sizes) such that any 4-bit or 5-bit number is composite, up to n around 200.

T. D. Noe, Table of n, a(n) for n=1..265

Cf. A081091, A081092.

for(n=2, 1000, f=0; for(i=1, n-1, t=2^n+2^i+1;  if(isprime(t), f=1; break)); if(!f, print1(n", ")))

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

cp's OEIS Frontend

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

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A360448 Indices of primes of the form p = 2^i + 2^j + 1, i > j > 0 (A081091).

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A014499 Number of 1's in binary representation of n-th prime.

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A014311 Numbers with exactly 3 ones in binary expansion.

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A039687 Primes of the form 3*2^k + 1.

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A081092 Primes having a prime number of 1's in their binary representation.

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Maple

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