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A087155 Primes having nontrivial palindromic representation in some (at least one) base.

%I A087155 #42 Feb 24 2023 16:01:31
%S A087155 5,7,13,17,23,29,31,37,41,43,59,61,67,71,73,83,89,97,101,107,109,113,
%T A087155 127,131,151,157,173,181,191,193,197,199,211,227,229,233,239,241,251,
%U A087155 257,271,277,281,307,313,331,337,349,353,373,379,383,397,401,409,419
%N A087155 Primes having nontrivial palindromic representation in some (at least one) base.
%C A087155 Number of terms < 10^n: 2, 18, 129, 1010, 8392, ..., . - _Robert G. Wilson v_, Jun 19 2014
%C A087155 Every whole number has single-digit representation in all large bases and all greater than 2 have representation 11 in the base one less than itself. Other palindromic representations are the nontrivial ones. - _James G. Merickel_, Jul 25 2015
%C A087155 Primes not in A016038. - _Robert Israel_, Jul 27 2015
%H A087155 Robert G. Wilson v, <a href="/A087155/b087155.txt">Table of n, a(n) for n = 1..10000</a>
%e A087155 17 is in the list because 17_2 = 10001 and 17_4 = 101, two nontrivial palindromic representations. 19 is not in the list because 19 is not a multidigit palindrome in any base other than base 18.
%p A087155 filter:= proc(n) local b,L;
%p A087155 if not isprime(n) then return false fi;
%p A087155 for b from 2 to floor(sqrt(n)) do
%p A087155   L:= convert(n,base,b);
%p A087155   if L = ListTools:-Reverse(L) then return true fi;
%p A087155 od:
%p A087155 false
%p A087155 end proc:
%p A087155 select(filter, [2*i+1 $ i=1..1000]); # _Robert Israel_, Jul 27 2015
%t A087155 palindromicBases[n_] := Module[{p}, Table[p = IntegerDigits[n, b]; If[p == Reverse[p], {b, p}, Sequence @@ {}], {b, 2, n - 2}]]; Select[ Prime@ Range@ 300, palindromicBases[#] !={}&] (* _Robert G. Wilson v_, May 06 2014 *)
%o A087155 (PARI) q=1; forprime(m=3,500,count=0; for(b=2,m-1, w=b+1; k=0; i=m; while(i>0,k=k*w+i%b; i=floor(i/b)); l=0; j=k; while(j>0,l=l*w+j%w; j=floor(j/w)); if(l==k,count=count+1; if(count>1,print1(m,", "); q=b; m=nextprime(m+1); q=1; b=1,q=b),)))
%Y A087155 Cf. A016038.
%K A087155 base,nonn
%O A087155 1,1
%A A087155 _Randy L. Ekl_, Oct 18 2003
%E A087155 Title, comments and example changed to agree with convention on single-digit numbers and incorporate 'nontrivial' concept by _James G. Merickel_, Jul 25 2015