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A379872 Numbers k that are the product of the lower half of their nontrivial divisors.

%I A379872 #30 Apr 02 2025 09:51:27
%S A379872 1,24,30,40,56,64,70,105,135,154,165,182,189,195,231,273,286,297,351,
%T A379872 357,374,385,399,418,429,442,455,459,494,513,561,595,598,621,627,646,
%U A379872 663,665,715,729,741,759,782,805,874,875,897,935,957,969,986,1001,1015,1023,1045,1054,1085
%N A379872 Numbers k that are the product of the lower half of their nontrivial divisors.
%C A379872 All terms under 1 million except 1, 2^6, 3^6, 5^6 and 7^6  have 6 nontrivial divisors, with p^6 for p prime having 5 nontrivial divisors, and so it seems that each term in the sequence is the product of three distinct numbers. - Edited by _Robert Israel_, Feb 04 2025
%C A379872 The majority of the terms are the product of 3 primes, but there are also terms of the form p*q*p^2, p*p^2*q, or p*p^2*p^3.
%C A379872 The first consecutive integers that appear in the sequence are a(45)=874 and a(46)=875.
%H A379872 Robert Israel, <a href="/A379872/b379872.txt">Table of n, a(n) for n = 1..10000</a>
%H A379872 Tom Gadron, <a href="/A379872/a379872.java.txt">Java Program</a>
%e A379872 24 is a term because the nontrivial divisors of 24 are 2,3,4,6,8,12, and 24=2*3*4.
%e A379872 30 is a term because the nontrivial divisors of 30 are 2,3,5,6,10,15, and 30=2*3*5.
%e A379872 135 is a term because the nontrivial divisors of 135 are 3,5,9,15,27,45, and 135=3*5*9.
%e A379872 729 is a term because the nontrivial divisors of 729 are 3,9,27,81,243, and 729=3*9*27.
%p A379872 isA379872 := proc(n)
%p A379872     local d;
%p A379872     numtheory[divisors](n) minus {1,n} ;
%p A379872     d := sort(convert(%,set)) ;
%p A379872     mul( op(i,d),i=1..floor((nops(d)+1)/2)) ;
%p A379872     if % = n then
%p A379872         true;
%p A379872     else
%p A379872         false;
%p A379872     end if;
%p A379872 end proc:
%p A379872 A379872 := proc(n)
%p A379872     option remember ;
%p A379872     if n =1 then
%p A379872         1;
%p A379872     else
%p A379872         for a from procname(n-1)+1 do
%p A379872             if isA379872(a) then
%p A379872                 return a;
%p A379872             end if;
%p A379872         end do:
%p A379872     end if;
%p A379872 end proc:
%p A379872 seq(A379872(n),n=1..60) ; # _R. J. Mathar_, Jan 29 2025
%t A379872 q[k_] := Times @@ Select[Divisors[k], #^2 <= k &] == k; Select[Range[1200], q] (* _Amiram Eldar_, Jan 05 2025 *)
%o A379872 (Java) \\ See Gadron link.
%o A379872 (PARI) isok(k) = my(d=divisors(k)); d=setminus(d, Set([1,k])); vecprod(Vec(d, #d\2 + #d%2)) == k; \\ _Michel Marcus_, Jan 05 2025
%Y A379872 Cf. A072499.
%K A379872 nonn
%O A379872 1,2
%A A379872 _Tom Gadron_, Jan 04 2025