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

1,2

Comments

Empirically, all terms under 1 million have 6 nontrivial divisors, with perfect squares having 5 nontrivial divisors, and so it seems that each term in the sequence is the product of three distinct numbers.

The majority of the terms are the product of 3 primes, but there are also terms in 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.

Links

Table of n, a(n) for n=1..57.

Tom Gadron, Java Program

Example

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.

Mathematica

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

Prog

(Java) \\ See Gadron link.
(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

Crossrefs

Cf. A072499.

Sequence in context: A030626 A302571 A335054 * A348604 A347063 A371920

Adjacent sequences: A379869 A379870 A379871 * A379873 A379874 A379875

Keyword

nonn,new

Author

Tom Gadron, Jan 04 2025

Status

approved