A392728 Numbers k whose smallest prime factor is equal to the number of distinct prime factors of k and whose largest prime factor is equal to the number of prime factors of k counted with multiplicity.

12, 18, 80, 200, 448, 500, 1250, 1568, 5488, 8505, 11264, 14175, 19208, 19845, 23625, 33075, 39375, 46305, 53248, 55125, 61952, 65625, 67228, 77175, 91875, 108045, 128625, 180075, 235298, 252105, 340736, 346112, 1082565, 1114112, 1515591, 1804275, 1874048, 2249728

Offset

1,1

Comments

Numbers k such that A020639(k) = A001221(k) and A006530(k) = A001222(k).

From Michael De Vlieger, Feb 07 2026: (Start)

This sequence contains nonsquarefree numbers that are not perfect powers, therefore A303946 is a superset.

The only strictly prime(i)-smooth numbers in this sequence are 12 and 18. Therefore a(n) for n > 2 is in A080259.

For n > 2, a(n) is in either A386434 if powerful (then, strictly Achilles) or A386294 if not powerful.

Even terms are of the form 2^k * p^m, with odd prime p = k+m, where k and m are coprime.

The smallest a(n) with A020639(a(n)) = prime(j) is prime(j)^(prime(j+prime(j)-1) - prime(j) + 1) * Product_{i=1..prime(j)-1} prime(i+j). Examples: 12 = 2^2*3, 8505 = 3^5 * 5 * 7, and 20772705078125 = 5^13 * 7 * 11 * 13 * 17. (End)

Links

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

Michael De Vlieger, Notes on this sequence.

Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n,i), n = 1..2048, 12X vertical exaggeration, with a color function where m = 1 is black, m = 2 is red, m = 3 is orange, ... m = 52 in magenta. The stripe below the plot indicates a(n) in A386294 with blue, and a(n) in A386434 with purple, showing 12 and 18 also in blue.

Example

80 = 2^4*5^1 is a term because its smallest prime factor equals the number of distinct prime factors (2), and its largest prime factor equals the number of prime factors counted with multiplicity (5).

Maple

A392728List := proc(N) # To get all terms <= N
    local s, t, c, u, q, k, r, i;
    r := [];
    for s from 2 to ilog2(N) do
        if isprime(s) then
            t := s;
            do
                t := nextprime(t);
                if N < s^t then break; end if;
                for c in combinat:-choose(select(isprime, [seq(i, i = s + 1 .. t - 1)]), s - 2) do
                    q := [s, op(c), t];
                    for u in combinat:-composition(t, s) do
                        k := mul(q[i]^u[i], i = 1 .. s);
                        if k <= N then r := [op(r), k]; end if;
                    end do;
                end do;
            end do;
        end if;
    end do;
    convert(sort(convert(r, set)), list);
end proc;
A392728List(2249728);

Mathematica

okQ[k_]:=PrimeNu[k]==FactorInteger[k][[1, 1]]&&PrimeOmega[k]==FactorInteger[k][[-1, 1]]; Select[Range[10^6], okQ] (* James C. McMahon, Feb 06 2026 *)

Prog

(PARI) isok(k) = if (k>1, my(f=factor(k)); (vecmin(f[, 1]) == omega(f)) && (vecmax(f[, 1]) == bigomega(f))); \\ Michel Marcus, Feb 07 2026

Crossrefs

Cf. A001221, A001222, A006530, A003586, A020639, A052486, A055932, A080259, A332785, A368089, A386294, A386434.

Supersets: A007916, A013929, A024619, A126706, A303946.

Sequence in context: A226176 A177426 A232384 * A086301 A231094 A258427

Adjacent sequences: A392725 A392726 A392727 * A392729 A392730 A392731

Keyword

nonn

Author

Felix Huber, Jan 30 2026

Status

approved