A052033 -id:A052033 - cp's OEIS frontend

A052026 Composites base 10 that remain composite in all bases b, 2<=b<=10, expansions interpreted as decimal numbers.

14, 18, 20, 24, 30, 32, 36, 40, 42, 44, 51, 54, 60, 62, 69, 70, 72, 74, 76, 78, 80, 86, 90, 92, 96, 98, 99, 100, 102, 104, 108, 110, 112, 114, 120, 124, 125, 126, 128, 129, 130, 132, 135, 140, 144, 146, 148, 150, 152, 156, 158, 159, 160, 162, 164, 168, 170, 174

Offset: 1

Patrick De Geest, Dec 15 1999

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 24. Primes in several bases, The Prime Puzzles & Problems Connection.

Cf. A038537, A052027-A052033, A084482, A236356

Select[Range@ 174, AllTrue[Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], CompositeQ] &] (* Michael De Vlieger, Mar 24 2015, version 10 *)

A038537 Primes base 10 that remain primes in eight bases b, 2<=b<=10, when the expansions are interpreted as decimal numbers.

Original entry on oeis.org

2, 3, 379081, 59771671, 146752831, 764479423, 1479830551, 3406187401, 5631714889, 7740024337, 8256310441, 8772257161, 9522879913, 10350894331, 12852250993, 14261996563, 16082349433, 16199980009, 17727606151, 18172964503, 18294784903, 19393314433, 19472325391, 20582035993

Offset: 1

Carlos Rivera

Sebastian Petzelberger, Table of n, a(n) for n = 1..54 (terms up to 10^12).
Carlos Rivera, Puzzle 24: Primes in several bases, The Prime Puzzles & Problems Connection.

Cf. A052026, A038537, A052027-A052033, A084482, A236356

Select[Prime[Range@ 50000], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], True] == 8 &] (* Michael De Vlieger, Mar 20 2015, after Harvey P. Dale at A052032 *)

a(4)-a(7) found by Jack Brennen (see link) added by Patrick De Geest, Dec 15 1999

Terms beyond a(7) from Sebastian Petzelberger, Mar 21 2015

A084482 Primes base 10 that remain primes in all nine bases b, 2<=b<=10, when the expansions are interpreted as decimal numbers.

Original entry on oeis.org

50006393431, 727533146383, 2250332130313, 2651541199513, 4437592255351, 4877749016143, 6777899690983, 7417899095713, 7431376081543, 7766799025303, 9078654198463, 10712216924641, 12244626455491, 13562282568103, 14180813918071, 14833027106593, 19479075240913, 19971686697103, 23196986067193, 34431442237963, 36429184518721, 49198998504223

Offset: 1

Jack Brennen, Jun 29 2003

a(1) found by Jack Brennen on Jul 13 2001; remaining terms computed by Jack Brennen, Nov 15 2001.

The number must end with 1, 3, 7, or 9 in each base from 2 to 10; thus must be congruent to: 1 (mod 2), 1 (mod 3), 1 or 3 (mod 4), 1 or 3 (mod 5), 1 (mod 6), 1 or 3 (mod 7), 1 or 3 or 7 (mod 8), 1 or 7 (mod 9), 1 or 3 or 7 or 9 (mod 10).

Carlos Rivera, PP and P Puzzle 24: Primes in several bases

Cf. A052026, A038537, A052027-A052033, A084482, A236356.

isok(n) = sum(b=2, 10, isprime(subst(Pol(digits(n, b)), x, 10))) == 9; \\ Michel Marcus, Mar 22 2015

Thanks to David W. Wilson for proposing the sequence and W. Edwin Clark for verifying the terms using Maple's command isprime.

A052027 Primes in base 10 that remain primes in seven bases b, 2<=b<=10, expansions interpreted as decimal numbers.

Original entry on oeis.org

5, 9241, 17791, 330289, 391231, 1005481, 1210483, 2378143, 2469241, 2779939, 2840041, 6817501, 8320831, 9865711, 10871407, 11087191, 12259603, 13645393, 15665833, 16707883, 17694463, 25751863, 27794287, 31488481

Offset: 1

Patrick De Geest, Dec 15 1999

Sebastian Petzelberger, Table of n, a(n) for n = 1..1000
Carlos Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A052026, A038537, A052027-A052033, A084482, A236356.

Select[Prime[Range@ 100000], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], True] == 7 &] (* Michael De Vlieger, Mar 21 2015, after Harvey P. Dale at A052032 *)

Missing terms 2378143 and 2469241 added by Sebastian Petzelberger, Mar 21 2015

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

A052032 Primes base 10 that remain prime in one (and only one) other base b, 2<=b<10, expansions interpreted as decimal numbers.

Original entry on oeis.org

41, 53, 73, 107, 113, 131, 137, 139, 167, 173, 223, 233, 239, 257, 271, 293, 317, 389, 401, 467, 491, 509, 521, 557, 593, 641, 661, 691, 701, 739, 761, 809, 827, 829, 839, 853, 859, 863, 881, 887, 911, 937, 971, 977, 991, 1013, 1063, 1109, 1129, 1151, 1153

Offset: 1

Patrick De Geest, Dec 15 1999

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000 [This replaces an earlier b-file computed by Harvey P. Dale]
C. Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A038537, A052026-A052033.

Select[Prime[Range[200]],Count[PrimeQ/@Table[FromDigits[ IntegerDigits[ #,i]],{i,2,9}],True]==1&] (* Harvey P. Dale, Oct 13 2012 *)

Definition clarified by Harvey P. Dale, Oct 13 2012

A052028 Primes base 10 that remain primes in six bases b, 2<=b<=10, expansions interpreted as decimal numbers.

Original entry on oeis.org

157, 523, 1249, 1483, 1753, 4051, 9187, 10531, 22921, 25981, 29599, 35899, 51031, 57751, 67579, 79939, 98323, 103561, 110581, 148471, 150193, 150343, 249703, 259183, 277063, 278623, 331081, 335833, 353401, 391903, 424819, 435553, 504547

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.

Select[Prime[Range@ 42000], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], True] == 6 &] (* Michael De Vlieger, Mar 20 2015, after Harvey P. Dale at A052032 *)

A256350 Composites in base 10 that remain composite in exactly eight bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

Original entry on oeis.org

4, 6, 12, 26, 27, 35, 38, 45, 46, 48, 49, 50, 52, 56, 57, 58, 63, 64, 65, 66, 68, 77, 81, 82, 84, 85, 88, 95, 105, 116, 117, 118, 119, 121, 122, 134, 136, 138, 142, 153, 154, 161, 165, 166, 171, 175, 176, 187, 188, 190, 192, 195, 207, 208, 218, 219, 220, 225

Offset: 1

Sebastian Petzelberger, Mar 25 2015

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Carlos Rivera, PP&P Puzzle 24: Primes in several bases, The Prime Puzzles & Problems Connection.

Cf. A052026, A256350-A256356, A038537, A052027-A052033, A084482, A236356.

A256356 Composites in base 10 that remain composite in exactly two bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

Original entry on oeis.org

33247243, 64037779, 104865433, 130237003, 238561081, 550677781, 947051353, 1013991553, 1246382791, 1343122201, 1607697631, 1609062751, 1632753601, 1788658063, 2203645111, 2364166213, 2393866411, 2480419783, 2518589671, 2544177511, 2668538575, 3029334883

Offset: 1

Sebastian Petzelberger, Mar 25 2015

Are there any remaining composites in only one other base?

Sebastian Petzelberger, Table of n, a(n) for n = 1..150
Carlos Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A052026, A256350, A256351, A256352, A256353, A256354, A256355.

Cf. A038537, A052027-A052033, A084482, A236356.

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

cp's OEIS Frontend

A052026 Composites base 10 that remain composite in all bases b, 2<=b<=10, expansions interpreted as decimal numbers.

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A038537 Primes base 10 that remain primes in eight bases b, 2<=b<=10, when the expansions are interpreted as decimal numbers.

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A084482 Primes base 10 that remain primes in all nine bases b, 2<=b<=10, when the expansions are interpreted as decimal numbers.

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A052027 Primes in base 10 that remain primes in seven bases 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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A052032 Primes base 10 that remain prime in one (and only one) other base b, 2<=b<10, expansions interpreted as decimal numbers.

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Mathematica

Extensions

A052028 Primes base 10 that remain primes in six bases b, 2<=b<=10, expansions interpreted as decimal numbers.

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Mathematica

A256350 Composites in base 10 that remain composite in exactly eight bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

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A256356 Composites in base 10 that remain composite in exactly two bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

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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.