login
Primes p whose base-8 expansion is also the decimal expansion of a prime.
5

%I #18 Mar 05 2022 11:08:59

%S 2,3,5,7,11,19,31,43,59,67,71,89,137,151,179,191,199,223,251,257,281,

%T 283,307,311,337,353,359,367,383,409,419,433,443,449,523,563,617,619,

%U 641,659,727,787,809,811,857,887,907,919,947,977,1033,1039,1097,1123

%N Primes p whose base-8 expansion is also the decimal expansion of a prime.

%C In general rebase notation (Marc LeBrun): p8 = (8) [p] (10).

%H Harry J. Smith, <a href="/A065726/b065726.txt">Table of n, a(n) for n = 1..1000</a>

%H M. F. Hasler, <a href="https://docs.google.com/document/d/10IM7fcAbB2tqRGuwfGvuEGUzD_IXbgXPDK0tfxN4M3o/pub">Primes whose base c expansion is also the base b expansion of a prime</a>

%e E.g., 787_10 = 1423_8 is prime, and so is 1423_10.

%t Select[ Range[2500], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 8]]] & ]

%o (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, 8); if (isprime(b), write("b065726.txt", n++, " ", p); if (n==1000, return)) ) } \\ _Harry J. Smith_, Oct 28 2009

%o (PARI) is(p, b=10, c=8)=isprime(vector(#d=digits(p, c), i, b^(#d-i))*d~)&&isprime(p) \\ This code can be used for other bases b, c when b>c. See A235265 for code also valid for b<c. \\ _M. F. Hasler_, Jan 12 2014

%Y Cf. A065720 up to A065727, A065361.

%Y Cf. A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

%K nonn,base

%O 1,1

%A _Patrick De Geest_, Nov 15 2001

%E Definition clarified by _M. F. Hasler_, Jan 12 2014