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

Author

Patrick De Geest, Nov 15 2001

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

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

Crossrefs

Cf. A065720-A065727, A065361.

Cf. A004678, A107715, A090707.

Programs

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]]] & ]

PARI

isok(p)={ isprime(p) && isprime(fromdigits(digits(p,4))) } \\ Harry J. Smith, Oct 27 2009

Formula

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

Extensions

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