A364023 Palindromes that have at least two distinct prime factors and whose prime factors, listed (with multiplicity) in descending order and concatenated, form a palindrome in base 10.

111, 414, 777, 35853, 1226221, 7673767, 7744477, 9396939, 859767958, 11211911211, 12467976421, 72709290727, 93969696939, 1030507050301, 1120237320211, 1225559555221, 1234469644321, 1334459544331, 3254595954523, 10048622684001, 100330272033001, 100827848728001

Offset: 1

Author

Vitaliy Kaurov, Jul 04 2023

Keywords

• nonn
• base
• hard

Examples

			111 = 37*3
414 = 23*3*3*2
777 = 37*7*3
35853 = 37*19*17*3
1226221 = 1201*1021
7673767 = 79111*97
7744477 = 3119*191*13
9396939 = 31013*101*3
859767958 = 2731*199*113*7*2
		

Crossrefs

Similar to A364050. Subsequence of A002113 and A024619.

Programs

Mathematica

(* generate palindromes with even n *)
poli[n_Integer?EvenQ]:=FromDigits[Join[#,Reverse[#]]]&/@
DeleteCases[Tuples[Range[0,9],n/2],{0..,_}]
(* generate palindromes with odd n *)
poli[n_Integer?OddQ]:=Flatten[Table[FromDigits[Join[#,{k},Reverse[#]]]&/@
DeleteCases[Tuples[Range[0,9],(n-1)/2],{0..,_}],{k,0,9}]]
(* find descending factor sequence *)
descendFACTOR[n_Integer]:=
PalindromeQ[StringJoin[Reverse[ToString/@Flatten[Table[#1,#2]&@@@#]]]]&&
Length[#]>1&@FactorInteger[n]
(* example for palindromes of size 7 *)
Parallelize@Select[poli[7],descendFACTOR]//Sort//AbsoluteTiming