A052294 -id:A052294 - cp's OEIS frontend

A084345 Numbers with a nonprime number of 1's in their binary expansion (complement of A052294).

0, 1, 2, 4, 8, 15, 16, 23, 27, 29, 30, 32, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 63, 64, 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, 128, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 159, 163

Offset: 1

Zak Seidov Jun 22 2003

			15 is in the sequence because 15_10=1111_2 and 1+1+1+1=4 is composite.

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

Cf. A052294.

Cf. A010051, A000120.

a084345 n = a084345_list !! (n-1)
a084345_list = filter ((== 0) . a010051' . a000120) [0..]
-- Reinhard Zumkeller, Aug 28 2013, Nov 16 2012

Select[Range[200],!PrimeQ[DigitCount[#,2,1]]&] (* Harvey P. Dale, Jan 21 2013 *)

for(n=0,200,b=binary(n); if(!isprime(sum(m=1,matsize(b)[2],b[m])),print1(n,",")))

More terms from Rick L. Shepherd, Jun 23 2003
Term 0 added by Michel Marcus, Aug 26 2013
b-file adjusted by Reinhard Zumkeller, Aug 28 2013

A363463 a(n) is the smallest number k with exactly n of its divisors in A052294.

Original entry on oeis.org

1, 3, 6, 12, 18, 48, 36, 192, 72, 84, 144, 3072, 168, 5985, 576, 336, 504, 26505, 672, 45045, 840, 1344, 6510, 129675, 2016, 1680, 11970, 4620, 4032, 389025, 3360, 888615, 6552, 13020, 53010, 6720, 8736, 855855, 90090, 23940, 13104, 2411955, 17472, 2417415, 26040

Offset: 0

Marius A. Burtea, Jul 08 2023

			a(0) = 1 because 1 has no divisors in A052294.
2 has no divisors in A052294 and 3 has only one divisor 3 = 11_2 in A052294, so a(1) = 3.
4 has no divisors in A052294, 5 has only the divisor 5 = 101_2 in A052294, 6 has divisors 3 = 11_2 and 6 = 110_2, so a(2) = 6.

Cf. A052294.

fp:=func; a:=[]; for n in [0..44] do k:= 1; while #[d:d in Divisors(k)|fp(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

seq[len_, kmax_] := Module[{s = Table[0, {len}], c = 0, k = 1, ind}, While[k < kmax && c < len, ind = DivisorSum[k, 1 &, PrimeQ[DigitCount[#, 2, 1]] &] + 1; If[ind <= len && s[[ind]] == 0, c++; s[[ind]] = k]; k++]; s]; seq[40, 10^6] (* Amiram Eldar, Jul 10 2023 *)

a(n) = my(k=1); while (sumdiv(k, d, isprime(hammingweight(d))) != n, k++); k; \\ Michel Marcus, Jul 10 2023

A363464 Numbers k in A052294 with arithmetic derivative k' (A003415) in A052294.

Original entry on oeis.org

6, 9, 10, 14, 18, 20, 21, 22, 24, 25, 33, 34, 35, 38, 40, 42, 44, 48, 49, 52, 62, 65, 66, 68, 69, 70, 76, 80, 84, 88, 91, 93, 94, 96, 100, 104, 110, 115, 117, 118, 121, 132, 133, 134, 138, 140, 143, 144, 145, 148, 152, 155, 158, 164, 174, 182, 185, 186, 188, 192

Offset: 1

Marius A. Burtea, Jul 08 2023

If p > 2 is in A092506 then m = 2*p and u = 4*p are terms. Indeed, if p = 2^k + 1, k >= 1, m = 2*(2^k + 1) = 2^(k+1) + 2^1 has two 1's in its binary expansion, and m' = p+2 = 2^k + 3 = 2^k + 2^1 + 1 has three 1's in its binary expansion. Similarly u = 4*(2^k + 1) = 2^(k+2) + 2^2 and u' = 4*p + 4 = 2^(k+2) + 2^3.
If p is in A057733 then the number m = 2*p is a term. Indeed, if p = 2^k + 3, k >= 1, m = 2*(2^k + 3) = 2^(k+1) + 2^2 + 2 has three 1's in its binary expansion, and m' = p+2 = 2^k + 5 = 2^k + 2^2 + 1 has three 1's in its binary expansion.
If p > 7 is in A057733 then the number m = 4*p is a term. Indeed, if p = 2^k + 3, k >= 3, m = 4*(2^k + 3) = 2^(k+2) + 2^3 + 2 has three 1's in its binary expansion, and m' = 4*(p + 1) = 4*(2^k + 4) = 2^(k+2) + 2^4 has two 1's in its binary expansion.
If p is in A123250 then the number m = 4*p is a term. Indeed, if p = 2^k + 5, k >= 1, m = 4*(2^k + 5) = 2^(k+2) + 2^4 + 2^2 has three 1's its binary expansion, and m' = 4*(p+1) = 4*(2^k + 6) = 2^(k+2) + 2^4 + 2^2 has three 1's in its binary expansion.
If p is in A104070 then the number m = 4*p is a term. Indeed, if p = 2^k + 9, k >= 1, m = 4*(2^k + 9) = 2^(k+2) + 2^5 + 2^2 has three 1's its binary expansion, and m' = 4*(p+1) = 4*(2^k + 10) = 2^(k+2) + 2^5 + 2^3 has three 1's in its binary expansion.

			6 = 110_2 has two 1's, 6' = 5 = 101_2 has two 1's, so 6 is a term.
9 = 101_2 has two 1's, 9' = 6 = 110_2 has two 1's, so 9 is a term.
10 = 1010_2 has two 1's, 10' = 7 = 111_2 has three 1's, so 10 is a term.
18 = 10010_2 has two 1's, 18' = 21 = 10101_2 has three 1's, so 18 is a term.

Cf. A000120, A003415, A052294, A057733, A081092, A092506, A104070, A123250.

fp:=func; f:=func; [n:n in [1..200]| fp(n) and fp(Floor(f(n)))];

pernQ[n_] := PrimeQ[DigitCount[n, 2, 1]]; d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[200], And @@ pernQ[{#, d[#]}] &] (* Amiram Eldar, Jul 10 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

A052467 Binomial transform of {b(n)}, where b(n)=1 for prime n and b(n)=0 otherwise.

Original entry on oeis.org

0, 1, 3, 6, 11, 20, 37, 70, 134, 255, 476, 869, 1564, 2821, 5201, 9948, 19793, 40562, 84271, 174952, 359576, 728805, 1457402, 2885051, 5681277, 11185110, 22103926, 43939533, 87864092, 176447165, 354929146, 713198803, 1428312446, 2846268351

Offset: 1

Eric W. Weisstein

nonn

Number of compositions of n into a prime number of parts. - Vladeta Jovovic, Jan 31 2005

The number of pernicious numbers (A052294) between 2^(n-1) and 2^n. Although the graph looks almost like 2^n, the graph of a(n)/2^n has quite a bit of variation. - T. D. Noe, Mar 14 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Binomial Transform.

Cf. A038499.

Cf. A010051, A121497.

b[n_] := Boole[ PrimeQ[n]]; a[n_] := Sum[ Binomial[n, k]*b[k], {k, 0, n}]; Table[a[n], {n, 0, 34}] // Differences (* Jean-François Alcover, Oct 25 2012 *)

G.f.: Sum_{k>=1} (x/(1 - x))^prime(k). - Ilya Gutkovskiy, Dec 28 2016

a(n) = A121497(n+1) - A121497(n). - Wesley Ivan Hurt, Jan 14 2022

More terms from David Wasserman, Feb 25 2002

Description corrected by T. D. Noe, May 17 2003

A144754 Integers that have a prime number of 0's in their binary expansion.

Original entry on oeis.org

4, 8, 9, 10, 12, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 32, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 56, 57, 58, 60, 65, 66, 68, 71, 72, 75, 77, 78, 79, 80, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 96, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110

Offset: 1

Leroy Quet, Sep 20 2008

			66 written in binary is 1000010. This has five 0's and five is a prime. So 66 is included in the sequence.

Indranil Ghosh, Table of n, a(n) for n = 1..25000

Cf. A052294.

Select[Range@ 120, PrimeQ@ DigitCount[#, 2, 0] &] (* Michael De Vlieger, Oct 26 2017 *)

isok(n) = isprime(#binary(n) - hammingweight(n)); \\ Michel Marcus, Feb 23 2016

from sympy import isprime
i=j=1
while j<=250:
    if isprime(bin(i)[2:].count("0")):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017

Many more terms from Reikku Kulon, Sep 21 2008

Name edited by Michel Marcus, Apr 30 2021

A121497 Binomial transform of the characteristic function of the prime numbers (A010051).

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 41, 78, 148, 282, 537, 1013, 1882, 3446, 6267, 11468, 21416, 41209, 81771, 166042, 340994, 700570, 1429375, 2886777, 5771828, 11453105, 22638215, 44742141, 88681674, 176545766, 352992931, 707922077, 1421120880, 2849433326

Offset: 0

T. D. Noe, Aug 03 2006

nonn

This is the binomial transform of the sequence {0,0,1,1,0,1,0,1,...}. Sequence A052467, the binomial transform of the sequence {0,1,1,0,1,0,1,...} is very similar. In fact, the first differences of this sequence yields A052467.
The number of pernicious numbers (A052294) less than 2^n. Although the graph looks almost like 2^n, the graph of a(n)/2^n has quite a bit of variation. - T. D. Noe, Mar 14 2009
a(n)/2^n is the probability that a series of Bernoulli trials with probability of success equal to 1/2 will result in a prime number of successes. Cf. A178851. - Eric M. Schmidt, Jul 13 2012
a(n) equals the number of subsets of [n] whose cardinalities are prime. - Ivan N. Ianakiev, Jul 14 2019
Upper and lower bounds are provided by Kim and Sinha (see links). - Jeffrey Shallit, Nov 14 2024

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 0..3324 (terms up to 1000 from Noe)
Sungjin Kim and Nilotpal Kanti Sinha, Binomial probability of prime number of successes, INTEGERS 20 (2020), #A99.
Vaclav Kotesovec, Plot of a(n) / (2^n/log(n/2)) for n = 2..10000

Cf. A000720, A010051, A052294, A052467, A178851

Primes:= select(isprime, [2,seq(i,i=3..100,2)]):
G:= add((z/(1-z))^p/(1-z),p=Primes):
S:= series(G,z,101):
seq(coeff(S,z,i),i=0..100); # Robert Israel, Sep 27 2018

Table[Sum[Binomial[n,Prime[i]], {i,PrimePi[n]}], {n,40}]

a(n)=my(s);forprime(p=2,n,s+=binomial(n,p));s \\ Charles R Greathouse IV, Mar 22 2013

a(n) = Sum_{i=1..pi(n)} binomial(n,prime(i)), where pi(n) is A000720(n), the number of primes <= n.
E.g.f.: exp(x) * (x^2/2! + x^3/3! + x^5/5! + ...) - Eric M. Schmidt, Jul 14 2012
G.f.: Sum_{p prime} x^p/(1-x)^(p+1). - Robert Israel, Sep 27 2018

a(0) inserted by Franklin T. Adams-Watters, Jul 13 2012

A280998 Numbers with a prime number of 1's in their binary reflected Gray code representation.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 23, 24, 25, 27, 28, 29, 30, 32, 33, 35, 37, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 60, 61, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 96, 97, 99, 101, 103

Offset: 1

Indranil Ghosh, Jan 12 2017

From Emeric Deutsch, Jan 28 2018: (Start)
Also the indices of the compositions that have a prime number of parts. For the definition of the index of a composition see A298644.
For example, 27 is in the sequence since its binary form is 11011 and the composition [2,1,2] has 3 parts.
On the other hand, 58 is not in the sequence since its binary form is 111010 and the composition [3,1,1,1] has 4 parts.
The command c(n) from the Maple program yields the composition having index n. (End)

			27 is in the sequence because the binary reflected Gray code representation of 27 is 10110 which has 3 1's, and 3 is prime.

Indranil Ghosh, Table of n, a(n) for n = 1..10001
Wikipedia, Gray code.

Cf. A014550, A052294, A005811, A298644, A101211.

Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]:
for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1:
r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc:
RunLengths := proc (L) map(nops, Runs(L)) end proc:
c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc:
A := {}: for n to 175 do if isprime(nops(c(n))) = true then A := `union`(A, {n}) else end if end do: A;
# most of the program is due to W. Edwin Clark. # Emeric Deutsch, Jan 28 2018

Select[Range[100], PrimeQ[DigitCount[BitXor[#, Floor[#/2]], 2, 1]] &] (* Amiram Eldar, May 01 2021 *)

is(n)=isprime(hammingweight(bitxor(n, n>>1))) \\ Charles R Greathouse IV, Jan 12 2017

A084561 Numbers with a square number of 1's in their binary expansion.

Original entry on oeis.org

0, 1, 2, 4, 8, 15, 16, 23, 27, 29, 30, 32, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 64, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 128, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178

Offset: 1

Jason Earls, Jun 27 2003

Begins to differ from A084345 at the 22nd term.

There are A003099(n) terms with at most n bits, so a(n) is n sqrt log n times a bounded function of n (which does not tend toward a limit). - Charles R Greathouse IV, Mar 26 2013

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

Cf. A001969, A000069, A052294, A084345.

Select[Range[0,178],IntegerQ[Sqrt[Count[IntegerDigits[#,2],1]]]&] (* Jayanta Basu, May 24 2013 *)

is(n)=issquare(hammingweight(n)) \\ Charles R Greathouse IV, Mar 26 2013

A355034 a(n) is the least base b >= 2 where the sum of digits of n is a prime number.

Original entry on oeis.org

3, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 3, 8, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 5, 2, 3, 3, 2, 4, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 4, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 4, 2, 6, 2, 2, 3, 18, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 6, 2, 2, 2, 2, 3, 2, 3, 4, 2, 2

Offset: 2

Rémy Sigrist, Jun 16 2022

The sequence is well defined:
- a(2) = 3,
- for n >= 3, the expansion of n in base n-1 is "11", with sum of digits 2.

			For n = 16:
- we have the following expansions and sum of digits:
     b  16_b     Sum of digits in base b
     -  -------  -----------------------
     2  "10000"                        1
     3    "121"                        4
     4    "100"                        1
     5     "31"                        4
     6     "24"                        6
     7     "22"                        4
     8     "20"                        2
- so a(16) = 8.

Rémy Sigrist, Table of n, a(n) for n = 2..10000

Cf. A052294, A216789, A355035 (corresponding prime numbers).

a(n) = for (b=2, oo, if (isprime(sumdigits(n,b)), return (b)))

from sympy import isprime
from sympy.ntheory.digits import digits
def a(n):
    b = 2
    while not isprime(sum(digits(n, b)[1:])): b += 1
    return b
print([a(n) for n in range(2, 89)]) # Michael S. Branicky, Jun 16 2022

a(n) = 2 iff n belongs to A052294.

a(n) <= n-1 for any n >= 3.

cp's OEIS Frontend

A084345 Numbers with a nonprime number of 1's in their binary expansion (complement of A052294).

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A363463 a(n) is the smallest number k with exactly n of its divisors in A052294.

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A363464 Numbers k in A052294 with arithmetic derivative k' (A003415) in A052294.

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

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A052467 Binomial transform of {b(n)}, where b(n)=1 for prime n and b(n)=0 otherwise.

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A144754 Integers that have a prime number of 0's in their binary expansion.

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A121497 Binomial transform of the characteristic function of the prime numbers (A010051).

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