A119378 - OEIS

%I #12 Nov 28 2022 01:47:24

%S 121,232,272,292,323,343,434,494,575,616,737,767,818,838,878,949,959,

%T 979,10201,10801,10901,11011,11611,11711,11911,12121,12221,12521,

%U 13031,13231,13531,13631,14041,14141,14641,14941,15151,15251,15751,15851,16261,16861,16961

%N Palindromic composites such that some digit permutation is prime.

%H Michael S. Branicky, <a href="/A119378/b119378.txt">Table of n, a(n) for n = 1..10000</a>

%e 121 is composite that have a prime digit permutation: 211.

%e 1001 is composite, but is not a term since 0011, though prime, contains leading zeros, which is not allowed here.

%t palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse@idn]; fQ[n_] := Union[PrimeQ /@ FromDigits /@ Permutations@ IntegerDigits@n][[ -1]] == True; Select[Range@15850, !PrimeQ@# && palQ[ #, 10] && fQ@# &] (* _Robert G. Wilson v_,  Aug 04 2006 *)

%o (Python)

%o from sympy import isprime

%o from itertools import count, islice, product

%o from sympy.utilities.iterables import multiset_permutations as mp

%o def pals(base=10): # generator for all palindromes as strings

%o   digits = "".join(str(i) for i in range(base))

%o   for d in count(1):

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

%o         if d//2 > 0 and p[0] == "0": continue

%o         left = "".join(p); right = left[::-1]

%o         for mid in [[""], digits][d%2]: yield left + mid + right

%o def ok(s): # where s is string of digits

%o     if isprime(int(s)): return False

%o     return any(p[0]!="0" and isprime(int("".join(p))) for p in mp(s))

%o def agen():

%o     yield from (int(s) for s in pals() if ok(s))

%o print(list(islice(agen(), 43))) # _Michael S. Branicky_, Nov 27 2022

%K base,nonn

%O 1,1

%A _Tanya Khovanova_, Jul 24 2006

%E More terms from _Joshua Zucker_ and _Robert G. Wilson v_, Aug 04 2006

%E a(41) and beyond from _Michael S. Branicky_, Nov 27 2022