A236437 -id:A236437 - cp's OEIS frontend

A052033 Primes base 10 that are never primes in any smaller base b, 2<=b<10, expansions interpreted as decimal numbers.

263, 269, 347, 397, 431, 461, 479, 499, 569, 599, 607, 677, 683, 719, 769, 797, 821, 929, 941, 1019, 1031, 1049, 1051, 1061, 1069, 1103, 1181, 1223, 1229, 1237, 1297, 1307, 1367, 1399, 1409, 1439, 1453, 1487, 1489, 1523, 1553, 1559, 1571, 1619, 1637

Offset: 1

Patrick De Geest, Dec 15 1999

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
C. Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A038537, A052026-A052033, A236174, A236437, A235354, A235377.

Select[Prime[Range@ 265], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 9}], True] == 0 &] (* Michael De Vlieger, Mar 20 2015 after Harvey P. Dale at A052032 *)

A235354 Minimal k > 1 such that the base-k representation of the n-th prime, read in decimal, is also prime.

Original entry on oeis.org

3, 2, 2, 4, 4, 4, 4, 4, 2, 4, 7, 4, 5, 4, 2, 4, 7, 4, 3, 4, 4, 3, 4, 2, 4, 2, 3, 4, 4, 4, 6, 4, 8, 3, 2, 4, 2, 2, 4, 2, 2, 3, 4, 3, 4, 2, 3, 8, 4, 2, 4, 7, 4, 4, 8, 10, 10, 9, 3, 5, 3, 4, 3, 4, 2, 4, 2, 6, 10, 3, 7, 4, 2, 3, 2, 2, 4, 10, 4, 3, 4, 3, 10, 3, 3

Offset: 1

Vladimir Shevelev, Jan 07 2014

Conjecture 1. Every number 2, ..., 10 occurs infinitely many times.
Conjecture 2. There exists limit of average (a(1) + ... + a(n))/n.
Conjecture: The average in Conjecture 2 exists and is equal to 10. - Charles R Greathouse IV, Jan 08 2014

			Prime(7) = 17. The base 2 representation of 17 is 10001, which reinterpreted in decimal is 73 * 137; the base 3 representation of 17 is 122, which reread as decimal is 2 * 61; and the base 4 representation of 17 is 101, which reread as decimal is prime, so therefore a(7) = 4.

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

Cf. A000040, A052033, A235377, A236174, A236437.

Table[Module[{b=2},While[!PrimeQ[FromDigits[IntegerDigits[p,b]]],b++];b],{p,Prime[Range[90]]}] (* Harvey P. Dale, Aug 30 2025 *)

rebase(n,from,to=10)=subst(Pol(digits(n,from)),'x,to)
a(n)=my(p=prime(n)); for(b=2,9,if(isprime(rebase(p,b)),return(b))); 10 \\ Charles R Greathouse IV, Jan 08 2014

More terms from Peter J. C. Moses

A236174 Maximal prime among the base-k representations of the n-th prime, read in decimal, for k=2,3,...,10.

Original entry on oeis.org

2, 11, 101, 13, 23, 31, 101, 103, 10111, 131, 43, 211, 131, 223, 101111, 311, 113, 331, 2111, 1013, 1021, 2221, 1103, 1011001, 1201, 1100101, 10211, 1223, 1231, 1301, 331, 2003, 211, 12011, 10010101, 2113, 10011101, 10100011, 2213, 10101101, 10110011, 20201, 2333, 21011, 3011, 11000111, 21211, 337, 3203, 11100101

Offset: 1

Vladimir Shevelev, Jan 19 2014

Let p = n-th prime. Write p in base k, k=2,3,4,5,..., and stop when the result is a prime when looked at in base 10. - N. J. A. Sloane, Jan 25 2014

			Let n=10, then prime(n)=29 (in base 10). The representations of 29 in bases 2,3,4,...,10 are 11101,1002,131,...,29 respectively. In this list 131 is the first and therefore the maximal prime. Thus a(10)=131.

Peter J. C. Moses, Table of n, a(n) for n = 1..2000

Cf. A000040, A052033, A235354, A235376, A235377, A236437.

Map[First[First[Select[Map[{#,PrimeQ[#]}&,Map[FromDigits,IntegerDigits[Prime[#],Range[2,10]]]],#[[2]]==True&]]]&,Range[50]]
Table[SelectFirst[Table[FromDigits[IntegerDigits[Prime[n],b]],{b,2,10}],PrimeQ],{n,80}] (* Harvey P. Dale, May 17 2024 *)

base_b(n, b) = {
  my(s=[], r, x);
  while(n>0,
    r = n%b;
    n = n\b;
    s = concat(r, s)
  );
  x=10;
  eval(Pol(s))
}
A236174(maxp) = {
  my(s=[], b, t);
  forprime(p=2, maxp,
    for(b=2, 10,
      t=base_b(p, b);
      if(isprime(t), s=concat(s, t); break)
    )
  );
  s
} \\ Colin Barker, Jan 23 2014

from sympy import prime, isprime
def A236174(n):
    p = prime(n)
    for b in range(2,11):
        x, y, z = p, 0, 1
        while x >= b:
            x, r = divmod(x,b)
            y += r*z
            z *= 10
        y += x*z
        if isprime(y):
            return y # Chai Wah Wu, Jan 03 2015

A235377 Positions of 10s in A235354.

Original entry on oeis.org

56, 57, 69, 78, 83, 89, 92, 95, 104, 109, 111, 123, 124, 128, 136, 139, 142, 158, 160, 171, 173, 176, 177, 178, 180, 185, 194, 200, 201, 203, 211, 214, 219, 222, 223, 228, 231, 236, 237, 241, 245, 246, 248, 256, 259, 270, 274, 280, 281, 296, 298, 302, 307, 314

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Jan 08 2014

If prime(a(n)) is written in base k>=2, and the k-representation is read in decimal, then all such numbers, for k = 2,3,...,9, are composite.

Cf. A052033, A235354, A235376, A236174, A236437.

isok(n) = {my(p = prime(n)); for (b = 2, 9, if (isprime(subst(Pol(digits(p, b)), x, 10)), return(0));); return (1);} \\ Michel Marcus, Jan 18 2014

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A052033 Primes base 10 that are never primes in any smaller base b, 2<=b<10, expansions interpreted as decimal numbers.

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A235354 Minimal k > 1 such that the base-k representation of the n-th prime, read in decimal, is also prime.

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A236174 Maximal prime among the base-k representations of the n-th prime, read in decimal, for k=2,3,...,10.

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A235377 Positions of 10s in A235354.

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