A236365 -id:A236365 - cp's OEIS frontend

A236537 Primes whose binary and ternary representations are also prime when read in decimal.

157, 199, 229, 313, 367, 523, 883, 1483, 2683, 2971, 3109, 3253, 3637, 4093, 4357, 4363, 4729, 4951, 5119, 5827, 6529, 9241, 10909, 11527, 13477, 15271, 15919, 18439, 19273, 19483, 22921, 24019, 29833, 31237, 31573, 32803, 35863, 35899, 36109, 36973, 39799

Offset: 1

K. D. Bajpai, Jan 28 2014

			157 is prime and appears in the sequence. Its representation in binary = 10011101 and in ternary = 12211 are also prime when read in decimal.
313 is prime and appears in the sequence. Its representation in binary = 100111001 and in ternary = 102121 are also prime when read in decimal.

K. D. Bajpai, Table of n, a(n) for n = 1..2115

Cf. A000040 (prime numbers), A065720 (primes: binary representation is also prime), A236365 (primes: binary and octal representation is also prime), A236512 (primes: base 2, 3, 4 and 5 representation are also prime).

t={}; n=1; While[Length[t] < 50, n=NextPrime[n]; If[PrimeQ[FromDigits[IntegerDigits[n,2]]] && PrimeQ[FromDigits[IntegerDigits[n,3]]], AppendTo[t,n]]]; t

base_b(n, b) = my(s=[], r, x=10); while(n>0, r = n%b; n = n\b; s = concat(r, s)); eval(Pol(s))
s=[]; forprime(p=2, 40000, if(isprime(base_b(p, 2)) && isprime(base_b(p, 3)), s=concat(s, p))); s \\ Colin Barker, Jan 28 2014

A236512 Primes whose representation in base (2), base (3), base (4) and base (5) are also prime when read in decimal.

Original entry on oeis.org

9241, 85303, 110581, 296011, 331081, 465523, 644353, 659371, 849943, 1108993, 1116163, 1210483, 2149471, 2469241, 2963923, 3409753, 3704203, 4451071, 4774801, 4978003, 5665213, 5674993, 5995021, 6507343, 6817501, 7529941, 7596373, 7693531, 7973653, 8320831, 8344681

Offset: 1

K. D. Bajpai, Jan 27 2014

			9241 is in the sequence because it is prime. Its representation in base (2):{10010000011001}, base (3):{110200021}, base (4):{2100121} and base (5):{243431}, when read in decimal are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..797

Cf. A000040 (prime numbers), A065720 (primes: binary representation is also prime),

A236365 (primes: binary and octal representation is also prime).

t={}; n=1; While[Length[t]<31,n=NextPrime[n]; If[PrimeQ[FromDigits[IntegerDigits[n,2]]]&&PrimeQ[FromDigits[IntegerDigits[n,3]]] &&PrimeQ[FromDigits[IntegerDigits[n,4]]]&&PrimeQ[FromDigits[IntegerDigits[n,5]]], AppendTo[t,n]]]; t

default(primelimit,2^31)
base_b(n, b) = {
  my(s=[], r, x=10);
  while(n>0,
    r = n%b;
    n = n\b;
    s = concat(r, s)
  );
  eval(Pol(s))
}
A236512(maxp) = {
  forprime(p=2, maxp,
    if(isprime(base_b(p, 2)) &&
       isprime(base_b(p, 3)) &&
       isprime(base_b(p, 4)) &&
       isprime(base_b(p, 5)), print1(p, ", ")
    )
  )
}
\\ Colin Barker, Jan 29 2014

A242677 Semiprimes whose binary representation, read in decimal, is also semiprime.

Original entry on oeis.org

15, 33, 55, 57, 65, 69, 77, 87, 115, 121, 129, 143, 169, 205, 209, 265, 299, 305, 321, 339, 361, 415, 417, 447, 451, 481, 493, 505, 517, 519, 535, 551, 553, 581, 611, 623, 667, 687, 695, 721, 737, 779, 789, 799, 865, 871, 893, 901, 905, 923, 943, 949, 955, 973

Offset: 1

K. D. Bajpai, May 20 2014

			a(2) = 33 = 3 * 11 is semiprime. Binary representation of 33 = 100001 = 11 * 9091 which is also semiprime.
a(4) = 57 = 3 * 19 is semiprime. Binary representation of 57 = 111001 = 11 * 10091 which is also semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001358, A236365, A236537, A241764.

with(numtheory):A242677 := proc() if bigomega(n)=2 and bigomega(convert(n, binary))=2 then RETURN (n) ; fi; end:  seq(A242677(), n=1..2000);

c = 0; Do[If [PrimeOmega[n] == 2 && PrimeOmega[FromDigits[IntegerDigits[n, 2]]] == 2, c ++;  Print[c, "  ", n]], {n, 1, 3*10^5}];

A279052 Semiprimes whose binary and ternary representations are prime when read in decimal.

Original entry on oeis.org

295, 1189, 2515, 4399, 4897, 5137, 7045, 7261, 7999, 8065, 9019, 9637, 10579, 10951, 10963, 11035, 11233, 12679, 13315, 13603, 13849, 16279, 18295, 20065, 20467, 20497, 23089, 23419, 23551, 23983, 26359, 27007, 27301, 27787, 29647, 33127, 33253, 33763, 34189, 34411

Offset: 1

K. D. Bajpai, Dec 05 2016

			295 is in the sequence because 295 = 5*59 (semiprime), 295_10 = 100100111_2 = 101221_3, and both 100100111_10 and 101221_10 are prime.
1189 is in the sequence because 1189 = 29*41 (semiprime), and both its binary representation 10010100101 and its ternary representation 1122001, if read as decimal numbers, are prime.

K. D. Bajpai and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 4075 terms from K. D. Bajpai)

Subsequence of A001358.

Cf. A000040, A007089, A036952, A065720, A236365, A236537.

Select[Range[50000], PrimeOmega[#] == 2 && PrimeQ[FromDigits[IntegerDigits[#, 2]]] && PrimeQ[FromDigits[IntegerDigits[#, 3]]] &]

has(n,b)=isprime(fromdigits(digits(n,b),10))
list(lim)=my(v=List(),t); forprime(p=2,lim\2, forprime(q=2,min(lim\p,p), if(has(t=p*q,2) && has(t,3), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Dec 05 2016

A281252 Numbers whose septenary, octal and nonary representations are prime when read in decimal.

Original entry on oeis.org

2, 3, 5, 43, 115, 619, 1249, 1681, 1711, 2563, 2635, 5155, 10321, 10531, 11539, 13219, 14479, 17713, 17755, 18217, 18889, 20203, 20905, 26335, 27163, 29305, 35353, 39859, 40867, 40897, 40993, 44425, 44803, 51145, 52993, 55735, 57751, 58075, 68335, 68839, 69553

Offset: 1

K. D. Bajpai, Jan 22 2017

After a(1) all the terms are odd.

After a(2) all terms are relatively prime to 42. - Charles R Greathouse IV, Jan 22 2017

			a(6) = 619 is in the sequence because 619_10 = 1543_7 = 1153_8 = 757_9; and 1543, 1153 and 757 are prime when read in decimal.
a(7) = 1249 is in the sequence because 1249_10 = 3433_7 = 2341_8 = 1637_9; and 3433, 2341 and 1637 are prime when read in decimal.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A065726, A065727, A138299, A175279, A236365.

Select[Range[500000], PrimeQ[FromDigits[IntegerDigits[#, 7]]] && PrimeQ[FromDigits[IntegerDigits[#, 8]]] && PrimeQ[FromDigits[IntegerDigits[#, 9]]] &]

is(n)=for(b=7,9, if(!isprime(fromdigits(digits(n,b))), return(0))); 1 \\ Charles R Greathouse IV, Jan 22 2017

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A236537 Primes whose binary and ternary representations are also prime when read in decimal.

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A236512 Primes whose representation in base (2), base (3), base (4) and base (5) are also prime when read in decimal.

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A242677 Semiprimes whose binary representation, read in decimal, is also semiprime.

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A279052 Semiprimes whose binary and ternary representations are prime when read in decimal.

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A281252 Numbers whose septenary, octal and nonary representations are prime when read in decimal.

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