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A364050 Palindromes that have at least two distinct prime factors and whose prime factors, listed (with multiplicity) in ascending order and concatenated, form a palindrome in base 10.

%I A364050 #42 Aug 05 2023 21:44:06
%S A364050 10001,36763,1037301,1226221,9396939,12467976421,14432823441,
%T A364050 93969696939,119092290911,1030507050301,1120237320211,1225559555221,
%U A364050 1234469644321,1334459544331,100330272033001,101222252222101,103023070320301,121363494363121,134312696213431
%N A364050 Palindromes that have at least two distinct prime factors and whose prime factors, listed (with multiplicity) in ascending order and concatenated, form a palindrome in base 10.
%C A364050 Palindromes p in A024619 such that A037276(p) is a palindrome.
%C A364050 Terms are coprime to 10. - _David A. Corneth_, Jul 05 2023
%e A364050   10001 = 73 * 137
%e A364050   36763 = 97 * 379
%e A364050 1037301 = 3 * 29 * 11923
%e A364050 1226221 = 1021 * 1201
%e A364050 9396939 = 3 * 101 * 31013
%t A364050 (* generate palindromes with even n *)
%t A364050 poli[n_Integer?EvenQ]:=FromDigits[Join[#,Reverse[#]]]&/@
%t A364050 DeleteCases[Tuples[Range[0,9],n/2],{0..,___}]
%t A364050 (* generate palindromes with odd n *)
%t A364050 poli[n_Integer?OddQ]:=Flatten[Table[FromDigits[Join[#,{k},Reverse[#]]]&/@
%t A364050 DeleteCases[Tuples[Range[0,9],(n-1)/2],{0..,___}],{k,0,9}]]
%t A364050 (* find ascending factor sequence *)
%t A364050 ascendFACTOR[n_Integer]:=
%t A364050 PalindromeQ[StringJoin[ToString/@Flatten[Table[#1,#2]&@@@#]]]&&
%t A364050 Length[#]>1&@FactorInteger[n]
%t A364050 (* example for palindromes of size 7 *)
%t A364050 Parallelize@Select[poli[7],ascendFACTOR]//Sort//AbsoluteTiming
%o A364050 (PARI) nextpal(n, b) = {my(m=n+1, p = 0); while (m > 0, m = m\b; p++; ); if (n+1 == b^p, p++); n = n\(b^(p\2))+1; m = n; n = n\(b^(p%2)); while (n > 0, m = m*b + n%b; n = n\b; ); m; }
%o A364050 ispal(n) = my(d=digits(n)); Vecrev(d) == d;
%o A364050 g(f) = my(s=""); for (i=1, #f~, for (j=1, f[i,2], s = concat(s, Str(f[i,1])))); eval(s);
%o A364050 isok(k) = my(f=factor(k)); if (#f~>=2, ispal(g(f)));
%o A364050 lista(nn) = {my(k=0); while (k <= nn, if (ispal(k) && isok(k), print1(k, ", ")); k = nextpal(k,10););} \\ _Michel Marcus_, Jul 11 2023
%Y A364050 Subsequence of A002113 and A024619. Cf. A037276.
%Y A364050 Similar to A364023.
%K A364050 nonn,base
%O A364050 1,1
%A A364050 _Vitaliy Kaurov_, Jul 03 2023