A081092 -id:A081092 - cp's OEIS frontend

A046704 Additive primes: sum of digits is a prime.

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131, 137, 139, 151, 157, 173, 179, 191, 193, 197, 199, 223, 227, 229, 241, 263, 269, 281, 283, 311, 313, 317, 331, 337, 353, 359, 373, 379, 397, 401, 409, 421, 443, 449, 461, 463, 467, 487, 557, 571, 577, 593

Offset: 1

Felice Russo

Sum_{a(n) < x} 1/a(n) is asymptotic to (3/2)*log(log(log(x))) as x -> infinity; see Harman (2012). Thus the sequence is infinite. - Jonathan Sondow, Jun 07 2012

Harman 2012 also shows, under a conjecture about primes in short intervals, that there are 3/2 * x/(log x log log x) terms up to x. - Charles R Greathouse IV, Nov 17 2014

			The digit sums of 11 and 13 are 1+1=2 and 1+3=4. Since 2 is prime and 4 is not, 11 is a member and 13 is not. - _Jonathan Sondow_, Jun 07 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Glyn Harman, Counting primes whose sum of digits is prime, J. Integer Seq., 15 (2012), Article 12.2.2.
Glyn Harman, Primes whose sum of digits is prime and metric number theory, Bull. Lond. Math. Soc. 44:5 (2012), pp. 1042-1049.

Indices of additive primes are in A075177.
Cf. A046703, A119450 = Primes with odd digit sum, A081092 = Primes with prime binary digit sum, A104213 = Primes with nonprime digit sum.
Cf. A007953, A010051; intersection of A028834 and A000040.

a046704 n = a046704_list !! (n-1)
a046704_list = filter ((== 1) . a010051 . a007953) a000040_list
-- Reinhard Zumkeller, Nov 13 2011

[ p: p in PrimesUpTo(600) | IsPrime(&+Intseq(p)) ];  // Bruno Berselli, Jul 08 2011

select(n -> isprime(n) and isprime(convert(convert(n,base,10),`+`)), [2,seq(2*i+1,i=1..1000)]); # Robert Israel, Nov 17 2014

Select[Prime[Range[100000]], PrimeQ[Apply[Plus, IntegerDigits[ # ]]]&]

isA046704(n)={local(s,m);s=0;m=n;while(m>0,s=s+m%10;m=floor(m/10));isprime(n) & isprime(s)} \\ Michael B. Porter, Oct 18 2009

is(n)=isprime(n) && isprime(sumdigits(n)) \\ Charles R Greathouse IV, Dec 26 2013

A027697 Odious primes: primes with odd number of 1's in binary expansion.

Original entry on oeis.org

2, 7, 11, 13, 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, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443, 457, 463, 487, 491, 499, 521, 541, 557, 563

Offset: 1

N. J. A. Sloane

Conjecture: a(n) < A027699(n) except for n = 2; verified up to n=5*10^7. Moreover, I conjecture that A027699(n) - a(n) tends to infinity. - Vladimir Shevelev

T. D. Noe, Table of n, a(n) for n=1..10000
E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, (French) [Sums of digits and almost primes] Math. Ann. 305 (1996), no. 3, 571--599. MR1397437 (97k:11029)
Ben Green, Three topics in additive prime number theory, arXiv:0710.0823 [math.NT], Oct 03, 2007, pp. 12-27.
Vladimir Shevelev, Generalized Newman phenomena and digit conjectures on primes, Internat. J. of Mathematics and Math. Sciences, 2008 (2008), Article ID 908045, 1-12.

Cf. A027699, A066148, A066149, A081092.

Cf. A000069 (odious numbers), A092246 (odd odious numbers)

a:=proc(n) local nn: nn:= convert(ithprime(n), base, 2): if `mod`(sum(nn[j], j =1..nops(nn)), 2)=1 then ithprime(n) else end if end proc: seq(a(n),n=1..103); # Emeric Deutsch, Oct 24 2007

Clear[BinSumOddQ];BinSumOddQ[a_]:=Module[{i,s=0},s=0;For[i=1,i<=Length[IntegerDigits[a,2]],s+=Extract[IntegerDigits[a,2],i];i++ ];OddQ[s]]; lst={};Do[p=Prime[n];If[BinSumOddQ[p],AppendTo[lst,p]],{n,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 06 2009 *)
Select[Prime@ Range@ 120, OddQ@ First@ DigitCount[#, 2] &] (* Michael De Vlieger, Feb 08 2016 *)

f(p)={v=binary(p);s=0;for(k=1,#v,if(v[k]==1,s++));return(s%2)};
forprime(p=2, 563, if(f(p), print1(p,", "))) \\ Washington Bomfim, Jan 14 2011

s=[]; forprime(p=2, 1000, if(norml2(binary(p))%2==1, s=concat(s, p))); s \\ Colin Barker, Feb 18 2014

from sympy import primerange
print([n for n in primerange(1, 1001) if bin(n)[2:].count("1")%2]) # Indranil Ghosh, May 03 2017

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A119480 Numbers n such that the Bernoulli number B_{4n} has denominator 30.

Original entry on oeis.org

1, 2, 17, 19, 31, 38, 47, 59, 61, 62, 71, 94, 101, 103, 107, 109, 118, 122, 137, 149, 151, 157, 167, 181, 197, 206, 211, 218, 223, 227, 229, 241, 257, 263, 269, 271, 283, 289, 302, 311, 313, 314, 317, 331, 334, 337, 347, 349, 353, 361, 362, 367, 379

Offset: 1

Alexander Adamchuk, Jul 26 2006

nonn

Most a(n) are primes from A043297(n) except for a(1) = 1 and composite a(n) for n=6,10,12,17,18,26,28,38,39,42,45,50,51, ... a(6) = 38 = 2*19, a(10) = 62 = 2*31, a(12) = 94 = 2*47, a(17) = 118 = 2*59, a(18) = 122 = 2*61, a(26) = 206 = 2*103, a(28) = 218 = 2*109, a(38) = 289 = 17*17, a(39) = 302 = 2*151, a(42) = 314 = 2*157, a(45) = 334 = 2*167, a(50) = 361 = 19*19, a(51) = 362 = 2*181, ... It appears that most composite a(n) are the doubles of some primes from A043297(n) belonging to A081092[n] and A045404[n] - Primes congruent to {3, 4, 5, 6} mod 7. The rest of composite a(n) are the squares of the primes from A043297(n).

Some a(n) are the products of different primes from A043297(n), for example a(77) = 527 = 17*31. a(n) belong to A045402 Primes congruent to {1, 3, 4, 5, 6} mod 7. a(n) is a subset of A053176 Primes p such that 2p+1 is composite, A045979 Bernoulli number B_{2n} has denominator 6, A090863 Numbers n such that F(n+1)*F(n-1)*B(2n) is an integer, where F(k)=k-th Fibonacci number and B(2k)=2k-th Bernoulli number. - Alexander Adamchuk, Jul 27 2006

E. Pérez Herrero, Table of n, a(n) for n=1..50000

Cf. A043297, A051225, A051222, A051230, A045404.

Cf. A045402, A053176, A045979, A090863.

Select[Range@ 400, Denominator@ BernoulliB[4 #] == 30 &] (* Michael De Vlieger, Aug 09 2017 *)

a(n) = A051225[n]/2.

A052294 Pernicious numbers: numbers with a prime number of 1's in their binary expansion.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 47, 48, 49, 50, 52, 55, 56, 59, 61, 62, 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

Jeremy Gow (jeremygo(AT)dai.ed.ac.uk), Feb 08 2000

No power of 2 is pernicious, but 2^n+1 always is.
If a prime p is of the form 2^k -1, then p is included in this sequence. - Leroy Quet, Sep 20 2008
There are A121497(n) n-bit members of this sequence. - Charles R Greathouse IV, Mar 22 2013
A list of programming codes for pernicious numbers can be found in the Rosetta Code link. - Martin Ettl, May 27 2014

			26 is in the sequence because the binary expansion of 26 is 11010 and 11010 has three 1's and 3 is prime, so the number of 1's in the binary expansion of 26 is prime. - _Omar E. Pol_, Apr 04 2016

Iain Fox, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..4000 from Daniel Arribas)
Rosetta Code, Pernicious numbers

Cf. A000069, A001969, A010051, A000120, A081092 (primes).

Cf. A262481 (subsequence).

a052294 n = a052294_list !! (n-1)
a052294_list = filter ((== 1) . a010051 . a000120) [1..]
-- Reinhard Zumkeller, Nov 16 2012

filter:= n -> isprime(convert(convert(n,base,2),`+`)):
select(filter, [$1..1000]); # Robert Israel, Oct 19 2014

Select[Range[6! ],PrimeQ[DigitCount[ #,2][[1]]]&] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2010 *)

is(n)=isprime(hammingweight(n)) \\ Charles R Greathouse IV, Mar 22 2013

from sympy import isprime
def ok(n): return isprime(bin(n).count("1"))
print([k for k in range(101) if ok(k)]) # Michael S. Branicky, Jun 16 2022

from sympy import isprime
def ok(n): return isprime(n.bit_count())
print([k for k in range(101) if ok(k)]) # Michael S. Branicky, Dec 27 2023

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.

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", ")))

A144214 Primes with both a prime number of 0's and a prime number of 1's in their binary representations.

Original entry on oeis.org

17, 19, 37, 41, 79, 103, 107, 109, 131, 137, 151, 157, 167, 173, 179, 181, 193, 199, 211, 227, 229, 233, 241, 257, 367, 379, 431, 439, 443, 463, 487, 491, 499, 521, 541, 557, 563, 569, 577, 587, 601, 607, 613, 617, 631, 641, 647, 653, 659, 661, 677, 701, 709

Offset: 1

Leroy Quet, Sep 14 2008

			79, a prime, in binary is 1001111. This has two 0's and has five 1's. Since both two and five are primes, 79 is included in the sequence.

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

Cf. A081092, A144213, A178350.

A080791 := proc(n) local i,dgs ; dgs := convert(n,base,2) ; nops(dgs)-add(i,i=dgs) ; end: A000120 := proc(n) local i,dgs ; dgs := convert(n,base,2) ; add(i,i=dgs) ; end: isA144214 := proc(n) local no0,no1 ; no0 := A080791(n) ; no1 := A000120(n) ; isprime(n) and isprime(no0) and isprime(no1) ; end: for n from 1 to 1200 do if isA144214(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 17 2008

Select[Prime[Range[6! ]],PrimeQ[DigitCount[ #,2,0]]&&PrimeQ[DigitCount[ #,2,1]]&] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2010 *)

from sympy import isprime
def ok(n): return isprime(c:=n.bit_count()) and isprime(n.bit_length()-c) and isprime(n)
print([k for k in range(710) if ok(k)]) # Michael S. Branicky, Dec 27 2023

More terms from R. J. Mathar, Sep 17 2008

A081093 a(n) is the smallest prime such that the number of 1's in its binary expansion is equal to the n-th prime.

Original entry on oeis.org

3, 7, 31, 127, 3583, 8191, 131071, 524287, 14680063, 1073479679, 2147483647, 266287972351, 4260607557631, 17591112302591, 246290604621823, 17996806323437567, 1152917106560335871, 2305843009213693951

Offset: 1

Reinhard Zumkeller, Mar 05 2003

a(n) = Min{p: A000120(p)=A000040(n), p prime}.
If 2^(Prime[n]) - 1 is a prime number, then a(n) = 2^(Prime[n]) - 1, where Prime[n] denotes the n-th prime number. This means that every Mersenne prime arises in this sequence. - Stefan Steinerberger, Jan 22 2006
For all n with prime(n) < 300, a(n) has either prime(n) or prime(n)+1 bits. - David Wasserman, Oct 25 2006

			n=4, p[4]=11, 3583=[11011111111] has 11 digits=1 and is prime;
2047=23.89=[11111111111] is not here because it is composite.
a(5)=3583=A081092(266)=A000040(502) having eleven 1's: '110111111111' and A000120(p)<11=prime(5) for primes p<3583.
Mersenne-primes are here, Mersenne composites not.

Cf. A000043, A000668, A001348, A061712, A000120, A014499.

Cf. A000040, A000120, A081092.

Do[k=1;While[Count[IntegerDigits[Prime[k], 2], 1] !=Prime[n], k++ ];Print[Prime[k]], {n, 1, 10}]

a(n) = A061712(A000040(n)). - Franklin T. Adams-Watters, Jun 06 2006

More terms from Franklin T. Adams-Watters, Jun 06 2006
Further terms from David Wasserman, Oct 25 2006
Edited by N. J. A. Sloane, Sep 15 2008 at the suggestion of R. J. Mathar

A085448 Primes having a semiprime number of 1's in their binary representation.

Original entry on oeis.org

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, 389, 401, 449, 461, 467, 523, 547, 571, 593, 599, 619, 643, 673, 683, 691, 739, 773, 797, 811, 821, 839, 853, 857, 881, 907, 937, 977

Offset: 1

Jason Earls, Aug 14 2003

Sequence of least prime with the number of 1's in its binary representation equal to n-th semiprime is: 23,311,991,2039,73727,63487,4128767,... What is the prime corresponding to 22?

			The prime 43 = '101011' has four 1's and so is a term.

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

Cf. A001358, A081092.

seqQ[n_] := PrimeQ[n] && PrimeOmega[DigitCount[n, 2, 1]] == 2; Select[Range[1000], seqQ] (* Amiram Eldar, Dec 14 2019 *)

A144213 Primes with a prime number of 0's in their binary representations.

Original entry on oeis.org

17, 19, 37, 41, 43, 53, 71, 79, 83, 89, 101, 103, 107, 109, 113, 131, 137, 151, 157, 167, 173, 179, 181, 193, 199, 211, 227, 229, 233, 241, 257, 263, 269, 277, 281, 293, 311, 317, 337, 347, 349, 353, 359, 367, 373, 379, 389, 401, 431, 439, 443, 449, 461, 463

Offset: 1

Leroy Quet, Sep 14 2008

			41, a prime, in binary is 101001. This has three 0's and 3 is prime, so 41 is in the sequence.

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

Cf. A081092, A144214. Intersection of A000040 and A144754.

A080791 := proc(n) local i,dgs ; dgs := convert(n,base,2) ; nops(dgs)-add(i,i=dgs) ; end: isA144213 := proc(n) local no0 ; no0 := A080791(n) ; if isprime(n) and isprime(no0) then true ; else false; fi; end: for n from 1 to 1200 do if isA144213(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 17 2008
# second Maple program:
q:= n-> isprime(n) and isprime(add(1-i, i=Bits[Split](n))):
select(q, [$1..500])[];  # Alois P. Heinz, Dec 27 2023

nmax = 100;
Select[Prime[Range[nmax]],
PrimeQ[Total@Mod[1 + IntegerDigits[#, 2], 2]] &] (* Andres Cicuttin, Jul 08 2020 *)
Select[Prime[Range[100]],PrimeQ[DigitCount[#,2,0]]&] (* Harvey P. Dale, Feb 03 2021 *)

from sympy import isprime
def ok(n): return isprime(n.bit_length()-n.bit_count()) and isprime(n)
print([k for k in range(464) if ok(k)]) # Michael S. Branicky, Dec 27 2023

More terms from R. J. Mathar, Sep 17 2008

cp's OEIS Frontend

A046704 Additive primes: sum of digits is a prime.

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A027697 Odious primes: primes with odd number of 1's in binary expansion.

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A119480 Numbers n such that the Bernoulli number B_{4n} has denominator 30.

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A052294 Pernicious numbers: numbers with a prime number of 1's in their binary expansion.

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

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

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