A390606 a(n) is the maximal number of perfect powers (not including 1) whose product is n^n, counting each perfect power as many times as it can be written with different exponents.

0, 1, 1, 2, 2, 4, 2, 6, 5, 6, 3, 9, 4, 8, 8, 11, 4, 13, 5, 13, 10, 10, 5, 18, 9, 12, 13, 16, 6, 21, 7, 20, 14, 14, 14, 24, 8, 16, 16, 25, 8, 24, 9, 23, 23, 18, 9, 32, 15, 24, 20, 25, 10, 31, 20, 31, 20, 20, 11, 39, 11, 22, 29, 35, 22, 33, 12, 30, 24, 36, 12, 44

Offset

1,4

Comments

a(n) is the largest number k such that Product_{i=1..k} A072103(x_i) = A000312(n) with x_i < x_i+1.

Links

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

Example

12^12 = 2^24*3^12. Write 2^24 as 2^2*2^3*2^4*4^2*2^5*2^6, with 2^6 replaceable by 4^3 or 8^2; write 3^12 as 3^2*3^3*3^7, or more generally 3^i*3^j*3^m with i + j + m = 12. In every case there are at most 9 perfect powers whose product is 12^12, hence a(12) = 9.

Maple

A390606 := proc(n)
    local F, a, t, m, k, b, d, f, u, T;
    T := table();
    u := proc(x) option remember; local d, f, t; d:=1; f:=ifactors(x)[2]; for t in f do d:=d*(t[2]+1) od; d end proc;
    F := ifactors(n)[2];
    a := 0;
    for t in F do
        m := n*t[2];
        for k from 2 do
            if assigned(T[k]) then b := T[k] else b := u(k)-1; T[k]:=b fi;
            if k*b < m-1 then
                a := a + b;
                m := m - k*b
            else
                a := a + iquo(m, k);
                break
            fi
        od
    od;
    a
end proc:
seq(A390606(n), n=1..72);

Crossrefs

Cf. A000005, A000312, A001597, A027750, A032741, A253642, A390604, A390605.

Sequence in context: A239240 A054929 A236628 * A078429 A384040 A360523

Adjacent sequences: A390603 A390604 A390605 * A390607 A390608 A390609

Keyword

nonn

Author

Felix Huber, Nov 14 2025

Status

approved