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

1, 24, 30, 40, 56, 64, 70, 105, 135, 154, 165, 182, 189, 195, 231, 273, 286, 297, 351, 357, 374, 385, 399, 418, 429, 442, 455, 459, 494, 513, 561, 595, 598, 621, 627, 646, 663, 665, 715, 729, 741, 759, 782, 805, 874, 875, 897, 935, 957, 969, 986, 1001, 1015, 1023, 1045, 1054, 1085

Offset: 1

Author

Tom Gadron, Jan 04 2025

Keywords

• nonn

Comments

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

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.

The first consecutive integers that appear in the sequence are a(45)=874 and a(46)=875.

Examples

			24 is a term because the nontrivial divisors of 24 are 2,3,4,6,8,12, and 24=2*3*4.
30 is a term because the nontrivial divisors of 30 are 2,3,5,6,10,15, and 30=2*3*5.
135 is a term because the nontrivial divisors of 135 are 3,5,9,15,27,45, and 135=3*5*9.
729 is a term because the nontrivial divisors of 729 are 3,9,27,81,243, and 729=3*9*27.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Tom Gadron, Java Program

Crossrefs

Cf. A072499.

Programs

Java

\\ See Gadron link.

Maple

isA379872 := proc(n)
    local d;
    numtheory[divisors](n) minus {1,n} ;
    d := sort(convert(%,set)) ;
    mul( op(i,d),i=1..floor((nops(d)+1)/2)) ;
    if % = n then
        true;
    else
        false;
    end if;
end proc:
A379872 := proc(n)
    option remember ;
    if n =1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA379872(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A379872(n),n=1..60) ; # R. J. Mathar, Jan 29 2025

Mathematica

q[k_] := Times @@ Select[Divisors[k], #^2 <= k &] == k; Select[Range[1200], q] (* Amiram Eldar, Jan 05 2025 *)

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