A344417 Number of palindromic factorizations of n.

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, 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, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset

1,4

Comments

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.

Links

Table of n, a(n) for n=1..87.

Formula

a(2^n) = A025065(n).

a(n) = A057567(A000188(n)). - Andrew Howroyd, May 22 2021

Example

The palindromic factorizations for n = 2, 4, 16, 36, 64, 144:
  (2)  (4)    (16)       (36)       (64)           (144)
       (2*2)  (4*4)      (6*6)      (8*8)          (12*12)
              (2*2*4)    (2*2*9)    (4*4*4)        (4*4*9)
              (2*2*2*2)  (3*3*4)    (2*2*16)       (4*6*6)
                         (2*2*3*3)  (2*2*4*4)      (2*2*36)
                                    (2*2*2*2*4)    (3*3*16)
                                    (2*2*2*2*2*2)  (2*2*6*6)
                                                   (3*3*4*4)
                                                   (2*2*2*2*9)
                                                   (2*2*3*3*4)
                                                   (2*2*2*2*3*3)

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
palQ[y_]:=Select[Permutations[y], #==Reverse[#]&]!={};
Table[Length[Select[facs[n], palQ]], {n, 50}]

Crossrefs

Positions of 1's are A005117.

The case of palindromic compositions is A016116.

The additive version (palindromic partitions) is A025065.

The case of palindromic prime signature is A242414.

The case of palindromic plane trees is A319436.

A001055 counts factorizations.

A229153 ranks non-palindromic partitions.

A265640 ranks palindromic partitions.

Cf. A000041, A000070, A000188, A004526, A030229, A056503, A057567, A082293, A344414.

Sequence in context: A335428 A368781 A050377 * A347437 A255231 A363265

Adjacent sequences: A344414 A344415 A344416 * A344418 A344419 A344420

Keyword

nonn

Author

Gus Wiseman, May 22 2021

Status

approved