A052294 -id:A052294 - cp's OEIS frontend

A144753 Positive integers whose binary representation is a palindrome and has a prime number of 1's.

3, 5, 7, 9, 17, 21, 31, 33, 65, 73, 93, 107, 127, 129, 257, 273, 313, 341, 381, 403, 443, 471, 513, 1025, 1057, 1137, 1193, 1273, 1317, 1397, 1453, 1571, 1651, 1707, 1831, 2047, 2049, 4097, 4161, 4321, 4433, 4593, 4681, 4841, 4953, 5189, 5349, 5461, 5709

Offset: 1

Leroy Quet, Sep 20 2008

Each term of this sequence is in both A006995 and A052294.

			21 in binary is 10101. This binary representation is a palindrome, it contains three 1's, and three is a prime. So 21 is a term.

Indranil Ghosh, Table of n, a(n) for n = 1..11167 (terms 1..1000 from Vincenzo Librandi)

Cf. A144752, A006995, A052294.

okQ[n_] := Module[{idn2 = IntegerDigits[n, 2]}, (idn2 == Reverse[idn2]) && PrimeQ[First[DigitCount[n, 2]]]];Select[Range[10000], okQ] (* Harvey P. Dale, Sep 23 2008 *)

from sympy import isprime
def ok(n): b = bin(n)[2:]; return b == b[::-1] and isprime(b.count("1"))
print(list(filter(ok, range(5710)))) # Michael S. Branicky, Sep 17 2021

# faster for computing initial segment of sequence
from sympy import isprime
from itertools import product
def ok2(bin_str): return isprime(bin_str.count("1"))
def bin_pals(maxdigits):
    yield from "01"
    digits, midrange = 2, [[""], ["0", "1"]]
    for digits in range(2, maxdigits+1):
        for p in product("01", repeat=digits//2-1):
            left = "1"+"".join(p)
            for middle in midrange[digits%2]:
                yield left + middle + left[::-1]
def auptopow2(e): return [int(b, 2) for b in filter(ok2, bin_pals(e))]
print(auptopow2(13)) # Michael S. Branicky, Sep 17 2021

More terms from Harvey P. Dale, Sep 23 2008

Name edited by Michael S. Branicky, Sep 17 2021

A262481 Numbers m having in binary representation exactly lpf(m) ones, where lpf = least prime factor = A020639; a(1) = 1.

Original entry on oeis.org

1, 6, 10, 12, 18, 20, 21, 24, 34, 36, 40, 48, 55, 66, 68, 69, 72, 80, 81, 96, 115, 130, 132, 136, 144, 155, 160, 185, 192, 205, 258, 260, 261, 264, 272, 273, 288, 295, 320, 321, 355, 384, 395, 425, 514, 516, 520, 528, 535, 544, 565, 576, 595, 623, 625, 637

Offset: 1

Reinhard Zumkeller, Sep 24 2015

			.   n | a(n) | A007088(a(n)) | factorization
. ----+------+---------------+--------------
.   1 |    1 |            1  |   1
.   2 |    6 |          110  |   2 * 3
.   3 |   10 |         1010  |   2 * 5
.   4 |   12 |         1100  |   2^2 * 3
.   5 |   18 |        10010  |   2 * 3^2
.   6 |   20 |        10100  |   2^2 * 5
.   7 |   21 |        10101  |   3 * 7
.   8 |   24 |        11000  |   2^3 * 3
.   9 |   34 |       100010  |   2 * 17
.  10 |   36 |       100100  |   2^2 * 3^2
.  11 |   40 |       101000  |   2^3 * 5
.  12 |   48 |       110000  |   2^4 * 3
.  13 |   55 |       110111  |   5 * 11
.  14 |   66 |      1000010  |   2 * 3 * 11
.  15 |   68 |      1000100  |   2^2 * 17
.  16 |   69 |      1000101  |   3 * 23
.  17 |   72 |      1001000  |   2^3 * 3^2
.  18 |   80 |      1010000  |   2^4 * 5
.  19 |   81 |      1010001  |   3^4
.  20 |   96 |      1100000  |   2^5 * 3
.  21 |  115 |      1110011  |   5 * 23
.  22 |  130 |     10000010  |   2 * 5 * 13
.  23 |  132 |     10000100  |   2^2 * 3 * 11
.  24 |  136 |     10001000  |   2^3 * 17
.  25 |  144 |     10010000  |   2^4 * 3^2  .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000120, A020639, A007088.

Subsequence of A052294.

a262481 n = a262481_list !! (n-1)
a262481_list = filter (\x -> a000120 x == a020639 x) [1..]

Select[Range[640], FactorInteger[#][[1, 1]] == DigitCount[#, 2, 1] &] (* Amiram Eldar, Jul 24 2023 *)

isok(n) = (n==1) || (hammingweight(n) == factor(n)[1,1]); \\ Michel Marcus, Sep 29 2015

A000120(a(n)) = A020639(a(n)).

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

A317294 Numbers with a noncomposite number of 1's in their binary expansion.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 47, 48, 49, 50, 52, 55, 56, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 79, 80, 81, 82, 84, 87, 88, 91, 93, 94, 96, 97, 98, 100

Offset: 1

Omar E. Pol, Aug 10 2018

Union of powers of 2 and pernicious numbers.
All powers of 2 are in the sequence because the binary expansion of a power of 2 contains only one digit "1" and 1 is a noncomposite number (A008578).
If k is in the sequence then so is 2*k. - David A. Corneth, Aug 10 2018

			8 is in the sequence because the binary expansion of 8 is 1000 and 1000 has one 1, and 1 is a noncomposite number (A008578).
26 is in the sequence because the binary expansion of 26 is 11010 and 11010 has three 1's, and 3 is a noncomposite number.

Robert Israel, Table of n, a(n) for n = 1..10000

Union of A000079 and A052294.
The complement is A317295.
All terms of A000051 are in this sequence.
Cf. A000069, A000120, A001969, A008578, A084345.

filter:= proc(n) local w;
  w:= convert(convert(n,base,2),`+`);
  w=1 or isprime(w)
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 15 2018

Select[Range[100], !CompositeQ[DigitCount[#, 2, 1]] &] (* Amiram Eldar, Jul 23 2023 *)

is(n) = my(h=hammingweight(n)); ispseudoprime(h) || h==1 \\ Felix Fröhlich, Aug 10 2018

A317295 Numbers with a composite number of 1's in their binary expansion.

Original entry on oeis.org

15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 63, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 95, 99, 101, 102, 105, 106, 108, 111, 113, 114, 116, 119, 120, 123, 125, 126, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 159, 163, 165, 166, 169, 170, 172, 175, 177, 178, 180, 183, 184, 187, 189, 190

Offset: 1

Omar E. Pol, Aug 10 2018

By definition no power of 2 is in the sequence.

			23 is in the sequence because the binary expansion of 23 is 10111 and 10111 has four 1's, and 4 is a composite number (A002808).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A317294.

Cf. A000069, A000120, A001969, A002808, A014312, A052294, A084345.

Select[Range[200], CompositeQ[DigitCount[#, 2, 1]] &] (* Amiram Eldar, Jul 23 2023 *)

isok(n) = my(w = hammingweight(n)); (w != 1) && !isprime(w); \\ Michel Marcus, Aug 15 2018

from sympy import isprime; isok = lambda n: n & (n-1) and not isprime(bin(n).count('1')) # David Radcliffe, Aug 15 2018

A343258 Numbers whose binary representation has a prime number of zeros and a prime number of ones.

Original entry on oeis.org

9, 10, 12, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 65, 66, 68, 72, 79, 80, 87, 91, 93, 94, 96, 103, 107, 109, 110, 115, 117, 118, 121, 122, 124, 131, 133, 134, 137, 138, 140, 143, 145, 146, 148, 151, 152, 155, 157, 158

Offset: 1

Jean-Jacques Vaudroz, Apr 09 2021

Terms of 4, 5 and 6 total bits (9 through 56) are the same as A089648.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 78 terms from Jean-Jacques Vaudroz)

Intersection of A052294 and A144754.

Cf. A089648.

q:= n->(l->(t->andmap(isprime, [t, nops(l)-t]))(add(i, i=l)))(Bits[Split](n)):
select(q, [$1..200])[];  # Alois P. Heinz, Apr 11 2021

Select[Range[160], And @@ PrimeQ[DigitCount[#, 2]] &] (* Amiram Eldar, Apr 09 2021 *)

isa(n)= isprime(hammingweight(n));
isb(n)= isprime(#binary(n) - hammingweight(n));
isok(n) = isa(n) && isb(n);

from sympy import isprime
def ok(n): b = bin(n)[2:]; return all(isprime(b.count(d)) for d in "01")
print(list(filter(ok, range(159)))) # Michael S. Branicky, Sep 10 2021

A355017 a(n) is the number of bases in 2..n in which the sum of the digits of n is prime.

Original entry on oeis.org

0, 1, 1, 3, 4, 4, 3, 5, 6, 7, 7, 8, 7, 8, 5, 11, 9, 10, 8, 13, 8, 12, 9, 13, 11, 12, 10, 15, 11, 16, 10, 17, 10, 20, 12, 20, 14, 18, 13, 21, 13, 22, 13, 20, 14, 25, 14, 22, 18, 22, 15, 26, 12, 29, 17, 25, 15, 27, 15, 30, 19, 26, 14, 32, 17, 33, 19, 27, 19, 31, 18, 34, 19, 29, 19, 37, 16, 33, 21, 30, 24, 39, 20, 38

Offset: 2

Samuel Harkness, Jun 15 2022

The graph of (n,a(n)) shows an interesting structure, somewhat resembling a comet with four tails. Starting at the bottom tail and going upwards:
Observations:
The bottom "tail" contains all n with both 2 and 3 as prime factors, i.e., numbers n in A008588 (1/6 of all n).
The second "tail" contains all n with 2 as a prime factor but not 3, i.e., numbers n in A047235 (1/3 of all n).
The third "tail" contains all n with 3 as a prime factor but 2, i.e., numbers n in A016945 (1/6 of all n).
The top "tail" contains all n with neither 2 nor 3 as a prime factor, i.e., numbers n in A007310 (1/3 of all n).
The bottom of each "tail" contains n with 5 as a prime factor. Moving up within each "tail," the prime factors of each n tend to increase.

			For n=7, express 7 in all bases from 2 to 7, then add the numbers, counting those which are prime:
  base 2: 1 1 1 --> 1+1+1=3 prime
  base 3: 2 1   --> 2+1=3   prime
  base 4: 1 3   --> 1+3=4   nonprime
  base 5: 1 2   --> 1+2=3   prime
  base 6: 1 1   --> 1+1=2   prime
  base 7: 1     --> 1=1     nonprime
The sum of the digits of the base-b expansion of 7 in 4 different bases b (2, 3, 5, and 6) from base 2 to 7 is prime, so a(7)=4.

Samuel Harkness, Table of n, a(n) for n = 2..10000
Samuel Harkness, Colored Scatterplot of the first 50000 terms, with n as multiples of 2 or 3
Samuel Harkness, Colored Scatterplot of the first 50000 terms, with the lowest factor of n among 5, 7, 11, or 13
Rémy Sigrist, Scatterplot of (n, b) such that the sum of digits of n in base b is prime (with n = 1..1000, b = 2..n).
Rémy Sigrist, Colored scatterplot of the first 50000 terms (where the color is function of gcd(n, 2*3*5)).

Cf. A008588, A047235, A016945, A007310, A028834, A052294.

a[n_] := Count[Range[2, n], ?(PrimeQ[Plus @@ IntegerDigits[n, #]] &)]; Array[a, 84, 2] (* _Amiram Eldar, Jun 17 2022 *)

a(n) = sum(b=2, n, isprime(sumdigits(n, b))); \\ Michel Marcus, Jun 16 2022

from sympy.ntheory import isprime, digits
def A355017(n): return sum(1 for b in range(2,n) if isprime(sum(digits(n,b)[1:]))) # Chai Wah Wu, Jun 17 2022

A380786 Numbers with a prime number of bits, prime number of ones, and prime number of zeros in their binary representation.

Original entry on oeis.org

17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 65, 66, 68, 72, 79, 80, 87, 91, 93, 94, 96, 103, 107, 109, 110, 115, 117, 118, 121, 122, 124, 4097, 4098, 4100, 4104, 4112, 4128, 4160, 4224, 4352, 4608, 5119, 5120, 5631, 5887, 6015, 6079, 6111, 6127, 6135, 6139, 6141, 6142, 6144, 6655, 6911

Offset: 1

Marc Morgenegg, Feb 03 2025

Each term has either two zeros or two ones in its binary representation. And so the total number of bits of each term is the larger prime of a twin prime pair.

			a(1) = 17 = 10001_2. Number of bits is 5, number of ones is 2, number of zeros is 3. {2,3,5} are all primes.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Intersection of A052294, A144754, and A380788.
Subsequence of primes gives A272478.
Subsequence of A343258.
Cf. A006512 (bitlengths of terms).

Select[Range[2^13], AllTrue[{#1, #2, #1 - #2} & @@ {IntegerLength[#,2], DigitCount[#, 2, 1]}, PrimeQ] & ] (* Michael De Vlieger, Feb 03 2025 *)

isok(k) = my(h=hammingweight(k), b=#binary(k)); isprime(h) && isprime(b) && isprime(b-h); \\ Michel Marcus, Feb 07 2025

from sympy import isprime
def ok(n): return isprime(L:=n.bit_length()) and isprime(O:=n.bit_count()) and isprime(L-O)
print([k for k in range(7160) if ok(k)]) # Michael S. Branicky, Feb 03 2025

from sympy import isprime, nextprime, sieve
from itertools import combinations, count, islice
def agen(): # generator of terms
    p = 5
    while True:
        passed = set()
        if isprime(p-2):
            for q in [2, p-2]:
                for locs in combinations(range(1, p), q-1):
                    w = ["1"] + ["0"]*(p-1)
                    for i in locs: w[i] = "1"
                    passed.add(int("".join(w), 2))
        yield from sorted(passed)
        p = nextprime(p)
        print(len(passed), p)
print(list(islice(agen(), 61))) # Michael S. Branicky, Feb 03 2025

A387305 Least k such that the Hamming weight (A000120) of n*k is prime.

Original entry on oeis.org

3, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 19, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 19, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 9, 3, 1, 1, 1, 1, 7, 1, 5, 3, 1, 1, 3, 3, 1, 19, 1, 1, 67, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 5, 3, 1, 1, 1, 1, 5, 1, 5, 3

Offset: 1

Pablo Cadena-Urzúa, Aug 25 2025

a(n) is always odd.

			a(1) = 3 because A000120(1) = 1 (not prime), A000120(2) = 1, and A000120(3) = 2 (prime).
a(11) = 1 because A000120(11) = 3 (prime).
a(15) = 19 since 15*19 = 285 and A000120(285) = 5 (prime); for 1 <= k < 19 the value A000120(15*k) is not prime.

Pablo Cadena-Urzúa, Table of n, a(n) for n = 1..10000

Cf. A000120, A052294.

A387305[n_] := Module[{k = -1}, While[!PrimeQ[DigitSum[(k += 2)*n, 2]]]; k];
Array[A387305, 100] (* Paolo Xausa, Sep 02 2025 *)

a(n) = {my(k=1); while(!isprime(hammingweight(n*k)), k++); k};
vector(100, n, a(n))  \\ first 100 terms

import sympy as sp
def a(n, kmax=10**6):
    for k in range(1, kmax + 1):
        if sp.isprime((n*k).bit_count()):
            return k
    return None
def A(N):
    return [a(n) for n in range(1, N + 1)]

a(n) = min{k >= 1 : A000120(n*k) is prime}.
a(n) = 1 if A000120(n) is prime (see A052294).
For all m >= 0, a(2^m) = 3.

A301797 a(n) = (4^prime(n) - 1)/3.

Original entry on oeis.org

5, 21, 341, 5461, 1398101, 22369621, 5726623061, 91625968981, 23456248059221, 96076792050570581, 1537228672809129301, 6296488643826193618261, 1611901092819505566274901, 25790417485112089060398421, 6602346876188694799461995861, 27043212804868893898596335048021

Offset: 1

André Dalwigk, Mar 26 2018

Also, numbers with a prime number of bits 1 (cf. A001348), interleaved with bits 0. Or: odd numbers with alternating binary digits and a prime Hamming weight A000120, cf. A052294. - M. F. Hasler, Oct 16 2018

			For n = 1, the first prime number is 2, so a(1) = (4^2-1)/3 = (16-1)/3 = 15/3 = 5;
for n = 2, prime(2) = 3, so a(2) = (4^3-1)/3 = (64-1)/3 = 63/3 = 21;
for n = 5, prime(5) = 11, so a(5) = (4^(11)-1)/3 = (4194304-1)/3 = 4194303/3 = 1398101.

Opinions on Puzzling.StackExchange about this sequence

Cf. A000040, A000120.
Subsequence of A006995 and of A002450, which is a subsequence of A000975.
Also the intersection of A002450 and A052294.

public static int a(int n){ int p = 1; while(n > 0){ p++; if(!new String(new char[p]).matches("(..+?)\\1+|.?")){ n--; } } return ((int) Math.pow(4, p)-1)/3; }

[(4^NthPrime(n) - 1)/3: n in [1..20]]; // Vincenzo Librandi, Oct 17 2018

a[n_]:=(4^Prime[n] - 1)/3; Array[a, 50] (* Stefano Spezia, Oct 16 2018 *)

a(n) = (4^prime(n) - 1)/3; \\ Michel Marcus, Mar 27 2018

a(n) = A002450(A000040(n)). - Michel Marcus, Mar 27 2018

cp's OEIS Frontend

A144753 Positive integers whose binary representation is a palindrome and has a prime number of 1's.

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A262481 Numbers m having in binary representation exactly lpf(m) ones, where lpf = least prime factor = A020639; a(1) = 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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A317294 Numbers with a noncomposite number of 1's in their binary expansion.

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A317295 Numbers with a composite number of 1's in their binary expansion.

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A343258 Numbers whose binary representation has a prime number of zeros and a prime number of ones.

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A355017 a(n) is the number of bases in 2..n in which the sum of the digits of n is prime.

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