A235463 Primes whose base-6 representation also is the base-2 representation of a prime.

7, 37, 43, 223, 1297, 1303, 1549, 7993, 9109, 46663, 54469, 55987, 326593, 1679659, 1681129, 1727569, 1734049, 1967587, 2006461, 2007763, 2014027, 2015287, 10077919, 10125649, 10125691, 10133467, 10412107, 10413397, 11757349, 11766421, 11766427, 11766637

Offset: 1

Author

M. F. Hasler, Jan 11 2014

Keywords

• nonn
• base

Comments

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=6, thus a subsequence of A077720.

Examples

			7 = 11_6 and 11_2 = 3 are both prime, so 7 is a term.
37 = 101_6 and 101_2 = 5 are both prime, so 37 is a term.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Crossrefs

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

Programs

Mathematica

  b62Q[n_]:=Module[{idn6=IntegerDigits[n,6]},Max[idn6]<2&&AllTrue[ {FromDigits[ idn6,6],FromDigits[idn6,2]},PrimeQ]]; Select[Prime[ Range[ 4,780000]],b62Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 29 2020 *)

PARI

is(p,b=2,c=6)=vecmax(d=digits(p,c))
				

PARI

forprime(p=1,1e3,is(p,6,2)&&print1(vector(#d=digits(p,2),i,6^(#d-i))*d~,",")) \\ To produce the terms, this is much more efficient than to select them using straightforwardly is(.)=is(.,2,6)