A235474 -id:A235474 - cp's OEIS frontend

A235615 Primes whose base-5 representation also is the base-4 representation of a prime.

2, 3, 13, 41, 43, 61, 181, 191, 263, 281, 283, 331, 383, 431, 443, 463, 641, 643, 661, 881, 911, 1063, 1091, 1291, 1303, 1531, 1693, 2083, 2143, 2203, 2293, 2341, 3163, 3181, 3191, 3253, 3343, 3593, 3761, 3931, 4001, 4093, 4391, 4691, 4793, 5011, 5393, 5413, 5441, 6301

Offset: 1

M. F. Hasler, Jan 13 2014

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

			Both 13 = 23_5 and 23_4 = 11 are prime.

Robert Price, Table of n, a(n) for n = 1..107268
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235474, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235639. See the LINK for further cross-references.

is(p,b=4,c=5)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,5,4)&&print1(vector(#d=digits(p,4),i,5^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,4,5)

A235473 Primes whose base-3 representation is also the base-4 representation of a prime.

Original entry on oeis.org

2, 43, 61, 67, 97, 103, 127, 139, 151, 157, 199, 211, 229, 277, 283, 331, 337, 349, 373, 379, 433, 439, 463, 499, 523, 571, 601, 607, 727, 751, 787, 823, 853, 883, 919, 991, 1063, 1087, 1117, 1213, 1249, 1327, 1381, 1429, 1483, 1531, 1567, 1597, 1627, 1759, 1783, 1867, 1999

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

This is a subsequence of A045331 and A045375.

			43 = 1121_3 and 1121_4 = 89 are both prime, so 43 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235266, A235474, A152079, A235475 - A235479, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,3],4]]&] (* Harvey P. Dale, Oct 16 2015 *)

is(p,b=4,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A359840 Numbers k that are the representation of primes in base 4 and in base 5.

Original entry on oeis.org

2, 3, 23, 131, 133, 221, 1211, 1231, 2023, 2111, 2113, 2311, 3013, 3211, 3233, 3323, 10031, 10033, 10121, 12011, 12121, 13223, 13331, 20131, 20203, 22111, 23233, 31313, 32033, 32303, 33133, 33331, 100123, 100211, 100231, 101003, 101333, 103333, 110021, 111211

Offset: 1

Bernard Schott, Jan 15 2023

For a(1) = 2, 2_4 = 2_5 = 2_10 and for a(2) = 3, 3_4 = 3_5 = 3_10; otherwise, these two primes are distinct for n >= 3 (example).

The corresponding sequences of primes are A235474 (for base 4) and A235615 (for base 5).

			a(3) = 23 because 23_4 = 11_10 = A235474(3) and 23_5 = 13_10 = A235615(3) are primes.
a(9) = 2023 because 2023_4 = 139_10 = A235474(9) and 2023_5 = 263_10 = A235615(9) are primes.

Intersection of A004678 and A004679.

Cf. A007090, A007091, A235474, A235615, A340290.

q[n_, b_] := Max[d = IntegerDigits[n]] < b && PrimeQ[FromDigits[d, b]]; Select[Range[200000], q[#, 4] && q[#, 5] &] (* Amiram Eldar, Jan 15 2023 *)

from sympy import isprime
def ok(n): return max(s:=str(n)) < '4' and isprime(int(s, 4)) and isprime(int(s, 5))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jan 15 2023

from sympy import isprime
from itertools import count, islice, product
def agen(): yield from (int(s) for d in count(1) for f in "123" for r in product("0123", repeat=d-1) if isprime(int(s:=f+"".join(r), 4)) and isprime(int(s, 5)))
print(list(islice(agen(), 40))) # Michael S. Branicky, Jan 15 2023

a(n) = A007090(A235474(n)); a(n) = A007091(A235615(n)).

More terms from Amiram Eldar, Jan 15 2023

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A235615 Primes whose base-5 representation also is the base-4 representation of a prime.

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A235473 Primes whose base-3 representation is also the base-4 representation of a prime.

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A359840 Numbers k that are the representation of primes in base 4 and in base 5.

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