A385998 Numbers that are divisible by an equal number of distinct primes and squares.

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 24, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 109, 113, 116, 117, 124, 127, 131, 135, 136, 137, 139, 147, 148, 149

Offset: 1

Author

Felix Huber, Aug 05 2025

Keywords

• nonn

Comments

The smallest term such that number of distinct primes = number of squares = k is:

k = 1: 2,

k = 2: 12,

k = 3: 240,

k = 4: 1260.

Examples

			12 is divisible by 2 distinct primes (2, 3) and by 2 squares (1, 4).
		

Links

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

Crossrefs

Supersequence of A000040.

Cf. A000290, A001221, A046951.

Programs

Maple

c:=(n,d)->igcd(n,d)=d and igcd(n/d,d)=d:
b:=n->nops(select(k->c(n,k),[seq(1..n)])):
A385998:=proc(n)
    option remember;
    local k;
    if n=1 then
        2
    else
        for k from procname(n-1)+1 do
            if b(k)=NumberTheory:-Omega(k,'distinct') then
                return k
            fi
        od
    fi;
end proc;	
seq(A385998(n),n=1..63);

Mathematica

q[k_] := Module[{e = FactorInteger[k][[;; , 2]]}, k > 1 && Length[e] == Times @@ (1 + Floor[e/2])]; Select[Range[150], q] (* Amiram Eldar, Aug 05 2025 *)

PARI

isok(m) = my(d=divisors(m)); #select(isprime, d) == #select(issquare, d); \\ Michel Marcus, Aug 05 2025