A065722 Primes that when written in base 4, then reinterpreted in base 10, again give primes.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 79, 83, 97, 103, 107, 109, 113, 131, 149, 151, 157, 163, 167, 181, 191, 193, 197, 227, 233, 241, 251, 277, 293, 307, 311, 313, 317, 349, 359, 373, 389, 401, 419, 421, 433, 443, 449, 463, 467, 503

Offset

1,1

Comments

In general rebase notation (Marc LeBrun): p4 = (4) [p] (10).

Links

Harry J. Smith, Table of n, a(n) for n=1..1000

Brady Haran and Grant Sanderson, Paterson Primes, YouTube/Numberphile video, Nov. 2022.

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

Formula

Numbers n such that A049084(n)>0 and A049084(A007090(n))>0. - R. J. Mathar, Jun 15 2007

Example

The decimal prime 29 in base 4 is 131 which is again a prime in base 10.
E.g., 509_10 = 13331_4 is prime but also 13331_10.

Maple

A007090 := proc(n) local b4digs ; b4digs := convert(n, base, 4) ; add( op(i, b4digs)*10^(i-1), i=1..nops(b4digs)) ; end: isA065722 := proc(n) local rebase ; if isprime(n) then rebase := A007090(n) ; RETURN(isprime(rebase)) ; else RETURN(false) ; fi ; end: for n from 1 to 1000 do p := ithprime(n) : if isA065722(p) then printf("%d, ", p) ; fi : od : # R. J. Mathar, Jun 15 2007

Mathematica

Select[ Range[505], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 4]]] & ]

Prog

(PARI) baseE(x, b)= { local(d, e=0, f=1); while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) } { n=0; for (m=1, 10^9, p=prime(m); b=baseE(p, 4); if (isprime(b), write("b065722.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 27 2009

Crossrefs

Cf. A065720-A065727, A065361.

Cf. A004678, A107715, A090707.

Sequence in context: A040177 A040184 A040988 * A335331 A279956 A197300

Adjacent sequences: A065719 A065720 A065721 * A065723 A065724 A065725

Keyword

nonn,base

Author

Patrick De Geest, Nov 15 2001

Extensions

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

Status

approved