A235477 -id:A235477 - cp's OEIS frontend

A262833 (7,2)-primes (defined in Comments).

2, 3, 5, 7, 17, 31, 43, 47, 61, 71, 73, 89, 101, 103, 113, 127, 157, 173, 197, 199, 229, 239, 241, 269, 271, 281, 311, 337, 353, 367, 379, 383, 397, 409, 439, 449, 463, 479, 509, 521, 577, 607, 617, 631, 647, 661, 673, 677, 719, 743, 761, 827, 857, 887, 911

Offset: 1

Clark Kimberling, Nov 01 2015

Let V = (b(1), b(2), ..., b(k)), where k > 1 and b(i) are distinct integers > 1 for j = 1..k. Call p a V-prime if the digits of p in base b(1) spell a prime in each of the bases b(2), ..., b(k).

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A000040, A262729, A262832, A262834.

{b1, b2} = {2, 7};
u = Select[Prime[Range[6000]], PrimeQ[FromDigits[IntegerDigits[#, b1], b2]] &]; (* A235477 *)
v = Select[Prime[Range[6000]], PrimeQ[FromDigits[IntegerDigits[#, b2], b1]] &]; (* A262833 *)
w = Intersection[u, v]; (* A262834 *)
(* Peter J. C. Moses, Sep 27 2015 *)

A262834 {2,7}-primes (defined in Comments).

Original entry on oeis.org

2, 31, 47, 103, 173, 199, 229, 367, 409, 463, 743, 827, 911, 967, 1123, 1163, 1321, 1447, 1583, 1657, 1669, 2161, 2647, 2677, 2861, 3361, 3673, 3851, 4079, 4231, 4271, 4513, 4663, 5003, 5381, 5923, 6329, 6569, 7043, 7103, 7393, 7561, 8263, 8753, 9649, 10337

Offset: 1

Clark Kimberling, Nov 05 2015

Let S = {b(1), b(2), ..., b(k)}, where k > 1 and b(i) are distinct integers > 1 for j = 1..k. Call p an S-prime if the digits of p in base b(i) spell a prime in each of the bases b(j) in S, for i = 1..k. Equivalently, p is an S-prime if p is a strong-V prime (defined at A262729) for every permutation of the vector V = (b(1), b(2), ..., b(k)).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A262729, A235266, A262833.

{b1, b2} = {2, 7};
u = Select[Prime[Range[6000]], PrimeQ[FromDigits[IntegerDigits[#, b1], b2]] &]; (* A235477 *)
v = Select[Prime[Range[6000]], PrimeQ[FromDigits[IntegerDigits[#, b2], b1]] &]; (* A262833 *)
w = Intersection[u, v]; (* A262834 *)
(* Peter J. C. Moses, Sep 27 2015 *)

A235464 Primes whose base-7 representation also is the base-2 representation of a prime.

Original entry on oeis.org

7, 2801, 17207, 19559, 134513, 134807, 840743, 842759, 842801, 941249, 943601, 958007, 958049, 958343, 5899657, 6591089, 6607903, 6706393, 6722857, 41196751, 41311663, 41314057, 46137673, 46137967, 46253257, 46942351, 46944409, 47059657

Offset: 1

M. F. Hasler, Jan 11 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.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
When the smaller base is b=2 such that only digits 0 and 1 are allowed, these are primes that are the sum of distinct powers of the larger base, here c=7, thus a subsequence of A077721.

			7 = 10_7 and 10_2 = 2 are both prime, so 7 is a term.
2801 = 11111_7 and 11111_2 = 31 are both prime, so 2801 is a term.

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235477, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

is(p,b=2,c=7)=vecmax(d=digits(p,c))

forprime(p=1,1e3,is(p,7,2)&&print1(vector(#d=digits(p,2),i,7^(#d-i))*d~,",")) \\ To produce the terms, this is much more efficient than to select them using straightforwardly is(.)=is(.,2,7)

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A262833 (7,2)-primes (defined in Comments).

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A262834 {2,7}-primes (defined in Comments).

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A235464 Primes whose base-7 representation also is the base-2 representation of a prime.

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