A392045 Odd positive integers whose canonical prime factorization p_1^e_1*...*p_r^e_r satisfies that {p_1, ..., p_r, e_1, ..., e_r} is a set of 2*r consecutive integers.

9, 81, 625, 2025, 5625, 15625, 117649, 455625, 1265625, 5764801, 22325625, 37515625, 43758225, 62015625, 73530625, 238239225, 337640625, 661775625, 25937424601, 45956640625, 90075015625, 246140015625, 482434430625, 683722265625, 2626587455625, 3138428376721

Offset

1,1

Comments

For r = 1, p_1 is an odd prime. For r > 1, p_(i+1) = p_i + 2. Every permutation of even numbers in [p_1 - 1; p_r - 1] and in [p_1 + 1; p_r + 1] gives a set of exponents e_i.

Links

Felix Huber, Table of n, a(n) for n = 1..222

Example

2025 = 3^4*5^2 is a term since 2025 is odd and 2, 3, 4, 5 are consecutive integers.
22325625 = 3^6*5^4*7^2 is a term since 22325625 is odd and 2, 3, 4, 5, 6, 7 are consecutive integers.

Maple

with(combinat):
A392045List := proc(N)
    local a, b, e, f, i, j, k, l, m, p, q;
    a := [];
    for i from 2 do
        b := [];
        q := ithprime(i);
        p := [];
        while isprime(q) do
            p := [op(p), q];
            q := q + 2;
        od;
        l := nops(p);
        for j to l do
            e := permute([seq(2*k, k = (p[1] - 1)/2 ... (p[j] - 1)/2)]);
            f := permute([seq(2*k, k = (p[1] + 1)/2 ... (p[j] + 1)/2)]);
            for m to nops(e) do
                b := [op(b), op(select(x -> x <= N, [mul(p[k]^e[m, k], k = 1 .. j), mul(p[k]^f[m, k], k = 1 .. j)]))];
            od
        od;
        if b = [] then
            break
        else
            a := [op(a), op(b)]
        fi
    od;
    sort(a)
end proc:
A392045List(3138428376721);

Crossrefs

Cf. A111774, A382330, A390784, A392046, A392047.

Cf. A097320, A383397, A387172, A387586, A389339, A389340, A389795.

Sequence in context: A213297 A272243 A272242 * A206728 A206857 A073531

Adjacent sequences: A392042 A392043 A392044 * A392046 A392047 A392048

Keyword

nonn,easy

Author

Felix Huber, Jan 05 2026

Status

approved