cp's OEIS Frontend

A344417 Number of palindromic factorizations of n.

%I A344417 #9 May 23 2021 00:20:08
%S A344417 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,4,1,1,
%T A344417 1,5,1,1,1,2,1,1,1,2,2,1,1,4,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,7,1,1,1,2,
%U A344417 1,1,1,5,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1
%N A344417 Number of palindromic factorizations of n.
%C A344417 A palindrome is a sequence that is the same whether it is read forward or in reverse. A palindromic factorization of n is a finite multiset of positive integers > 1 with product n that can be permuted into a palindrome.
%F A344417 a(2^n) = A025065(n).
%F A344417 a(n) = A057567(A000188(n)). - _Andrew Howroyd_, May 22 2021
%e A344417 The palindromic factorizations for n = 2, 4, 16, 36, 64, 144:
%e A344417   (2)  (4)    (16)       (36)       (64)           (144)
%e A344417        (2*2)  (4*4)      (6*6)      (8*8)          (12*12)
%e A344417               (2*2*4)    (2*2*9)    (4*4*4)        (4*4*9)
%e A344417               (2*2*2*2)  (3*3*4)    (2*2*16)       (4*6*6)
%e A344417                          (2*2*3*3)  (2*2*4*4)      (2*2*36)
%e A344417                                     (2*2*2*2*4)    (3*3*16)
%e A344417                                     (2*2*2*2*2*2)  (2*2*6*6)
%e A344417                                                    (3*3*4*4)
%e A344417                                                    (2*2*2*2*9)
%e A344417                                                    (2*2*3*3*4)
%e A344417                                                    (2*2*2*2*3*3)
%t A344417 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A344417 palQ[y_]:=Select[Permutations[y],#==Reverse[#]&]!={};
%t A344417 Table[Length[Select[facs[n],palQ]],{n,50}]
%Y A344417 Positions of 1's are A005117.
%Y A344417 The case of palindromic compositions is A016116.
%Y A344417 The additive version (palindromic partitions) is A025065.
%Y A344417 The case of palindromic prime signature is A242414.
%Y A344417 The case of palindromic plane trees is A319436.
%Y A344417 A001055 counts factorizations.
%Y A344417 A229153 ranks non-palindromic partitions.
%Y A344417 A265640 ranks palindromic partitions.
%Y A344417 Cf. A000041, A000070, A000188, A004526, A030229, A056503, A057567, A082293, A344414.
%K A344417 nonn
%O A344417 1,4
%A A344417 _Gus Wiseman_, May 22 2021