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A046477 Primes that are palindromic in bases 8 and 10. 0

%I #37 May 25 2023 13:25:36

%S 2,3,5,7,373,13331,30103,1496941,1970791

%N Primes that are palindromic in bases 8 and 10.

%C Any other terms have more than 20 digits. - _Michael S. Branicky_, Dec 19 2020

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/palpri.htm">World!Of Palindromic Primes</a>

%e 373_10 = 565_8. - _Jon E. Schoenfield_, Apr 10 2021

%t Do[s = RealDigits[n, 8][[1]]; t = RealDigits[n, 10][[1]]; If[PrimeQ[n], If[FromDigits[t] == FromDigits[Reverse[t]], If[FromDigits[s] == FromDigits[Reverse[s]], Print[n]]]], {n, 1, 10^5}]

%t pal810Q[p_]:=PalindromeQ[p]&&IntegerDigits[p,8]==Reverse[IntegerDigits[p,8]]; Select[ Prime[ Range[150000]],pal810Q] (* _Harvey P. Dale_, May 25 2023 *)

%o (Python) # efficiently search to large numbers

%o from sympy import isprime

%o from itertools import product

%o def candidate_prime_pals(digits):

%o ruled_out = "024568" # can't be even or multiple of 5

%o midrange = [[""], "0123456789"]

%o for p in product("0123456789", repeat=digits//2):

%o left = "".join(p)

%o if len(left):

%o if left[0] in ruled_out: continue

%o for middle in midrange[digits%2]:

%o yield left+middle+left[::-1]

%o for digits in range(1, 15):

%o for p in candidate_prime_pals(digits):

%o intp = int(p); octp = oct(intp)[2:]

%o if octp==octp[::-1]:

%o if isprime(intp):

%o print(intp, end=", ") # _Michael S. Branicky_, Dec 19 2020

%o (Python) # alternate sufficient for producing terms through a(9)

%o from sympy import isprime

%o def ispal(n): strn = str(n); return strn==strn[::-1]

%o for n in range(10**7):

%o if ispal(n) and ispal(oct(n)[2:]) and isprime(n):

%o print(n) # _Michael S. Branicky_, Dec 20 2020

%o (PARI) is(n) = my(d=digits(n, 8), dd=digits(n)); d==Vecrev(d) && dd==Vecrev(dd)

%o forprime(p=1, , if(is(p), print1(p, ", "))) \\ _Felix Fröhlich_, Dec 20 2020

%Y Cf. A002113, A002385, A029976, A029804.

%K nonn,hard,base,more

%O 1,1

%A _Patrick De Geest_, Aug 15 1998

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