A392046 Even 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.

2, 8, 48, 162, 2000, 2592, 3888, 5000, 25920, 58320, 120000, 750000, 911250, 2531250, 13720000, 19208000, 37340352, 56010528, 58320000, 85750000, 87127488, 87480000, 162000000, 196036848, 235298000, 300125000, 304946208, 405000000, 457419312, 546750000, 588245000

Offset

1,1

Comments

Because 2 must be a prime factor and, among any run of consecutive integers, the number of composites cannot be greater than the number of primes, the largest allowable prime factor in any term is 13, and the largest allowable exponent is 12. Consequently, the admissible prime sets p_i are {2, 3, 5, 7, 11}, {2, 3, 5, 7, 11, 13}, and {2} together with any selection from {3, 5, 7}, excluding {2, 7}. This yields a finite sequence of 920 terms in total. The largest term is 2^4*3^6*5^8*7^9*11^10*13^12 < 10^42.

Links

Felix Huber, Table of n, a(n) for n = 1..920

Example

162 = 2^1*3^4 is a term since 162 is even and 1, 2, 3, 4 are consecutive integers.
2000 = 2^4*5^3 is a term since 2000 is even and 2, 3, 4, 5 are consecutive integers.

Maple

with(combinat):
A392046List := proc(N)
    local a, c, d, e, i, j, k, l, p, s, t;
    a := [];
    p := [op(select(x -> x <> [2, 7], map(x -> [2, op(x)], choose([3, 5, 7])))), [2, 3, 5, 7, 11], [2, 3, 5, 7, 11, 13]];
    for i in p do
        l := nops(i);
        c := select(x -> not member(x, i), [seq(j, j = 2 .. i[l])]);
        d := l - nops(c);
        s := [[], [1], [1, i[l] + 1]][d + 1];
        t := [[], [i[l] + 1], [i[l] + 1, i[l] + 2]][d + 1];
        e := permute([op(s), op(c)]);
        if nops(t) = 0 then
            for j in e do
                a := [op(a), mul(i[k]^j[k], k = 1 .. l)]
            od
        else
            for j to nops(e) do
                a:=[op(a), mul(i[k]^e[j, k], k = 1 .. l), mul(i[k]^permute([op(c), op(t)])[j, k], k = 1 .. l)]
            od
        fi
    od;
    select(x -> x <= N, sort(a))
end proc:
A392046List(10^42);

Crossrefs

Cf. A111774, A382330, A390784, A392045, A392047.

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

Sequence in context: A058928 A376029 A228288 * A356346 A387139 A292277

Adjacent sequences: A392043 A392044 A392045 * A392047 A392048 A392049

Keyword

nonn,easy,fini,full

Author

Felix Huber, Jan 05 2026

Status

approved