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Nonpalindromic primes whose binary expansion, interpreted as a base-10 number, yields a palindromic prime.
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%I #21 Feb 11 2024 13:19:02

%S 443,6827,7607,19801,23581,31183,85093,97213,314777,364621,370477,

%T 382813,450011,524287,1077697,1159601,1177073,1215017,1299833,1311749,

%U 1356197,1458253,1547069,1589123,1613987,1649299,1716619,1851271,1893607,2092799,4404833,4454369,4671857

%N Nonpalindromic primes whose binary expansion, interpreted as a base-10 number, yields a palindromic prime.

%e 443 is a nonpalindromic prime. Its binary expansion is 110111011, which, when interpreted as a base-10 number, is a palindromic prime.

%t Select[Range[5000000], PrimeQ[#] && ! PalindromeQ[#] && PrimeQ[FromDigits[IntegerDigits[#, 2]]] && PalindromeQ[FromDigits[IntegerDigits[#, 2]]] &]

%t ppQ[p_]:=With[{c=FromDigits[IntegerDigits[p,2],10]},PrimeQ[c]&&PalindromeQ[c]]; Select[Prime[ Range[ 330000]],!PalindromeQ[#]&&ppQ[#]&] (* _Harvey P. Dale_, Feb 11 2024 *)

%o (Python)

%o from sympy import isprime, primerange

%o def ispal(s): return s == s[::-1]

%o def aupto(limit):

%o alst = []

%o for p in primerange(13, limit+1):

%o if not ispal(str(p)):

%o b = bin(p)[2:]

%o if ispal(b) and isprime(int(b)): alst.append(p)

%o return alst

%o print(aupto(5*10**6)) # _Michael S. Branicky_, Jun 13 2021

%Y Cf. A002385, A006995.

%K nonn,base

%O 1,1

%A _Tanya Khovanova_, Jun 13 2021