A144214 -id:A144214 - cp's OEIS frontend

A178350 Semiprimes with both a prime number of 0's and a prime number of 1's in their binary representations.

9, 10, 21, 22, 25, 26, 35, 38, 49, 65, 87, 91, 93, 94, 115, 118, 121, 122, 133, 134, 143, 145, 146, 155, 158, 161, 185, 194, 203, 205, 206, 213, 214, 217, 218, 319, 381, 382, 415, 445, 446, 471, 478, 493, 501, 502, 505, 515, 517, 527, 529, 535, 542, 545, 551

Offset: 1

Juri-Stepan Gerasimov, May 25 2010

			a(1)=9 because 9(written in base 10)=1001 where 2=prime number of 0's and 2=prime number of 1's.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A144214, A178064, A178065.

A000120 := proc(n) add(d,d=convert(n,base,2)) ; end proc:
A080791 := proc(n) dgs :=convert(n,base,2) ; nops(dgs)-A000120(n) ; end proc:
for n from 1 to 300 do spr :=A001358(n) ; if isprime( A080791(spr) ) and isprime(A000120(spr)) then printf("%d,",spr) ; end if; end do: # R. J. Mathar, Aug 12 2010

Select[Range[600],PrimeOmega[#]==2&&AllTrue[{DigitCount[ #,2,0], DigitCount[ #,2,1]},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 25 2016 *)

Corrected (167 removed) by R. J. Mathar, Aug 12 2010

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

A169817 n-th prime with both a prime number of 0's and a prime number of 1's in binary representation minus n-th semiprime with both a prime number of 0's and a prime number of 1's in their binary representation.

Original entry on oeis.org

8, 9, 16, 19, 54, 77, 72, 71, 82, 72, 64, 66, 74, 79, 64, 63, 72, 77, 78, 93, 86, 88, 95, 102, 209, 218, 246, 245, 240, 258, 281, 278, 285, 304, 323, 238, 182, 187, 162, 142, 155, 136, 135, 124, 130, 139, 142, 138, 142, 134, 148, 166, 167, 174, 176, 168, 177, 174

Offset: 1

Juri-Stepan Gerasimov, May 25 2010

nonn

			a(1)=A144214(1)-A178350(1)=17-9=8.

Cf. A014499, A035103, A178064, A178065.

pn0Q[n_]:=PrimeQ[DigitCount[n,2,1]]&&PrimeQ[DigitCount[n,2,0]]; nn=600;Module[{ps=Select[Prime[Range[nn]],pn0Q],sps=Select[Range[nn], PrimeOmega[#]==2&&pn0Q[#]&],minlen},minlen=Min[Length[ps], Length[ sps]];First[#]-Last[#]&/@Thread[{Take[ps,minlen],Take[sps,minlen]}]] (* Harvey P. Dale, May 07 2012 *)

a(n)=A144214(n)-A178350(n).

Corrected (96 replaced by 86, all numbers from a(27) on replaced) by R. J. Mathar, Jun 04 2010

A173347 Fibonacci numbers with both a prime number of 0's and a prime number of 1's in their binary representations.

Original entry on oeis.org

21, 233, 196418, 9227465, 165580141, 2971215073, 53316291173, 2504730781961, 3416454622906707, 51680708854858323072, 184551825793033096366333, 898923707008479989274290850145, 3210056809456107725247980776292056

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 16 2010

21 -> 10101, 233 -> 11101001, 196418 -> 101111111101000010,..

Cf. A000045, A144214.

Select[Fibonacci[Range[6! ]],PrimeQ[DigitCount[ #,2,0]]&&PrimeQ[DigitCount[ #,2,1]]&]

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A178350 Semiprimes with both a prime number of 0's and a prime number of 1's in their binary representations.

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

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A169817 n-th prime with both a prime number of 0's and a prime number of 1's in binary representation minus n-th semiprime with both a prime number of 0's and a prime number of 1's in their binary representation.

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A173347 Fibonacci numbers with both a prime number of 0's and a prime number of 1's in their binary representations.

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