A236174 -id:A236174 - cp's OEIS frontend

A236437 Primes which occur in their proper place in A236174.

2, 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, 1733, 1759, 1811, 1823, 1949, 1973, 1997

Offset: 1

N. J. A. Sloane, Jan 25 2014

Primes p such that A236174(k) = prime(k) for some k. The values of k are (essentially) given in A235377.

Same as A052033 if the initial 2 is omitted.

			263 is the 56th prime and is also the 56th term in A236174.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A052033, A236174, A235377, A235354.

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
A236437_list = [prime(n) for n in range(1,10**6) if A236174(n) == prime(n)]
# Chai Wah Wu, Jan 03 2015

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

Original entry on oeis.org

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

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

A253549 Maximal prime written in decimal among the base-k representations of the n-th prime, read in base 16, for k=2,3,...,16.

Original entry on oeis.org

2, 17, 257, 19, 19, 19, 65537, 37, 65809, 53, 307, 257, 53, 547, 563, 293, 101, 277, 4099, 577, 4129, 8737, 787, 137, 577, 257, 593, 4643, 4657, 773, 577, 821, 311, 313, 268501249, 74017, 74257, 8707, 359, 8753, 8963, 613, 9011, 12289, 285212929, 577, 135697

Offset: 1

Chai Wah Wu, Jan 03 2015

A base 16 variant of A236174.

			For n = 19, 67 is the 19th prime, and written in base 2, ..., is '1000011', '2111', '1003', '232', '151', '124', '103', '74', '67', '61', '57', '52', '4b', '47', '43'.  Out of these, when read in as hexadecimal numbers, the first prime is 1003_16 which is 4099_10.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A236174.

from sympy import prime, isprime
def A253549(n):
    p = prime(n)
    for b in range(2,17):
        x, y, z = p, 0, 1
        while x >= b:
            x, r = divmod(x,b)
            y += r*z
            z *= 16
        y += x*z
        if isprime(y):
            return y

cp's OEIS Frontend

A236437 Primes which occur in their proper place in A236174.

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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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A235377 Positions of 10s in A235354.

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PARI

A253549 Maximal prime written in decimal among the base-k representations of the n-th prime, read in base 16, for k=2,3,...,16.

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Python