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

%I A385998 #9 Aug 12 2025 18:47:06
%S A385998 2,3,5,7,11,12,13,17,18,19,20,23,24,28,29,31,37,40,41,43,44,45,47,50,
%T A385998 52,53,54,56,59,61,63,67,68,71,73,75,76,79,83,88,89,92,97,98,99,101,
%U A385998 103,104,107,109,113,116,117,124,127,131,135,136,137,139,147,148,149
%N A385998 Numbers that are divisible by an equal number of distinct primes and squares.
%C A385998 The smallest term such that number of distinct primes = number of squares = k is:
%C A385998   k = 1: 2,
%C A385998   k = 2: 12,
%C A385998   k = 3: 240,
%C A385998   k = 4: 1260.
%H A385998 Felix Huber, <a href="/A385998/b385998.txt">Table of n, a(n) for n = 1..10000</a>
%e A385998 12 is divisible by 2 distinct primes (2, 3) and by 2 squares (1, 4).
%p A385998 c:=(n,d)->igcd(n,d)=d and igcd(n/d,d)=d:
%p A385998 b:=n->nops(select(k->c(n,k),[seq(1..n)])):
%p A385998 A385998:=proc(n)
%p A385998     option remember;
%p A385998     local k;
%p A385998     if n=1 then
%p A385998         2
%p A385998     else
%p A385998         for k from procname(n-1)+1 do
%p A385998             if b(k)=NumberTheory:-Omega(k,'distinct') then
%p A385998                 return k
%p A385998             fi
%p A385998         od
%p A385998     fi;
%p A385998 end proc;	
%p A385998 seq(A385998(n),n=1..63);
%t A385998 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 *)
%o A385998 (PARI) isok(m) = my(d=divisors(m)); #select(isprime, d) == #select(issquare, d); \\ _Michel Marcus_, Aug 05 2025
%Y A385998 Supersequence of A000040.
%Y A385998 Cf. A000290, A001221, A046951.
%K A385998 nonn
%O A385998 1,1
%A A385998 _Felix Huber_, Aug 05 2025