A384204 a(n) is the number of palindromic permutations of the prime factors of n (counted with multiplicity).

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0

Offset

1,36

Comments

a(m) = 0 if and only if, in the canonical prime factorization of m, more than one prime has an odd exponent. Equivalently, a(m) > 0 if and only if m is either a perfect square or the product of a perfect square and a prime. A265640 gives the numbers m for which a(m) > 0.

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Formula

a(q*s^2) = a(s^2) = (Sum_{i=1..k} e_i)!/(Product_{i=1..k} e_i!), where q is prime, s is a positive integer and p_1^e_1*...*p_k^e_k is the canonical prime factorization of s.

a(m) = 0 if m is not of the form q*s^2 or s^2.

Example

a(1) = 1 because 1 has an empty factorization, [], which also is a palindrome.
a(72) = a(36) = 2 because 72 = 2*3*2*3*2 = 3*2*2*2*3 and 36 = 2*3*3*2 = 3*2*2*3.
a(144) = 3 because 144 = 2*2*3*3*2*2 = 2*3*2*2*3*2 = 3*2*2*2*2*3.
a(169) = 1 because 169 = 13*13.

Maple

A384204:=proc(n)
    local i, l;
    l:=[seq(i[2], i in ifactors(n)[2])];
    if nops(select(x->x mod 2<>0, l))>1 then
        0
    else
        l:=map(x->floor(x/2), l);
        return add(l)!/(mul(i!, i in l))
    fi;
end proc;
seq(A384204(n), n=1..88);

Crossrefs

Cf. A242414, A265640, A388859.

Sequence in context: A127325 A368750 A259660 * A119842 A015624 A015114

Adjacent sequences: A384201 A384202 A384203 * A384205 A384206 A384207

Keyword

nonn,easy

Author

Felix Huber, Sep 28 2025

Status

approved