A392195 Positive integers with canonical prime factorization Product_{i=1..m} p_i^(p_i - p_{i-1}), where p_0 = 1.

1, 2, 6, 9, 150, 225, 250, 625, 7350, 11025, 12250, 14406, 21609, 30625, 33614, 117649, 107611350, 161417025, 179352250, 210918246, 265734150, 316377369, 398601225, 442890250, 448380625, 492142574, 1107225625, 1286153286, 1722499009, 1929229929, 4715895382, 18186318150

Offset

1,2

Comments

If k = Product_{i=1..m} p_i^e_i is in the sequence, then p_i = 1 + Sum_{j<=i} e_j. Thus the primes in the factorization are obtained as cumulative sums of the exponents plus 1.

The only prime term is 2, the only squarefree terms are 1, 2 and 6.

Links

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

Formula

A001222(a(n)) = A006530(a(n)) - 1.

Example

150 = 2^1*3^1*5^2 is a term since 2 - 1 = 1 (exponent of 2), 3 - 2 = 1 (exponent of 3), and 5 - 3 = 2 (exponent of 5).
21609 = 3^2*7^4 is a term since 3 - 1 = 2 (exponent of 3), and 7 - 3 = 4 (exponent of 7).

Maple

A392195 := proc(N)
    local c, i, f, m, t;
    m := evalf(ln(N));
    t := table();
    c := 0;
    f := proc(l, v, k)
        local i, p, w;
        c := c + 1;
        t[c] := v;
        i := k;
        do
            p := ithprime(i);
            if (p - l)*evalf(ln(p)) > m then
                break
            end if;
            w := v*p^(p - l);
            if w <= N then
                f(p, w, i + 1)
            end if;
            i := i + 1;
        end do
    end proc;
    f(1, 1, 1);
    sort([seq(t[i], i = 1 .. c)]);
end proc:
A392195(18186318150);

Crossrefs

Cf. A001222, A006530, A390784, A392819.

Sequence in context: A342317 A117541 A244060 * A333185 A236106 A095105

Adjacent sequences: A392192 A392193 A392194 * A392196 A392197 A392198

Keyword

nonn

Author

Felix Huber, Mar 11 2026

Status

approved