A094519 Numbers having at least one pair (x,y) of divisors with x

6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246

Offset: 1

Author

Reinhard Zumkeller, May 06 2004

Keywords

• nonn

Comments

If m is in the sequence then so is k*m for k > 0. Furthermore, all terms are even. - David A. Corneth, Aug 31 2019

If (x,y) = (1,m) with m > 1, then oblong numbers m*(m+1) >= 6 belong to this sequence, and each oblong number >= 6 is a primitive term of the subsequence {k*m*(m+1), k >= 1}. Examples: with pair (1,2), we get multiples of 6 (see A008588); with (1,3) we get multiples of 12 (see A008594); with (1,4) we get multiples of 20 (see A008602); with (1,7) we get multiples of 56. - Bernard Schott, Aug 31 2019

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 22, 230, 2317, 23201, 232209, 2322920, 23232166, 232332309, 2323370184, ... . Apparently, the asymptotic density of this sequence exists and equals 0.23233... . - Amiram Eldar, Apr 20 2025

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A094518.

Complement of A094520.

A superset of A088723. - R. J. Mathar, Sep 16 2007

Subsequences: A002378 \ {0, 2}, A008588 \ {0}, A008602 \ {0}.

Programs

Mathematica

aQ[n_] := AnyTrue[Total /@ Subsets[Divisors[n], {2}], Divisible[n, #] &]; Select[Range[250], aQ] (* Amiram Eldar, Aug 31 2019 *)

PARI

is(n) = {my(d = divisors(n)); for(i = 1, #d - 2, for(j = i + 1, #d - 1, if(n % (d[i] + d[j]) == 0, return(1) ) ) ); 0 } \\ David A. Corneth, Aug 31 2019

Python

from itertools import count, islice
from sympy import divisors
def A094519_gen(): # generator of terms
    for n in count(1):
        for i in range(1,len(d:=divisors(n))):
            di = d[i]
            for j in range(i):
                if n % (di+d[j]) == 0:
                    yield n
                    break
            else:
                continue
            break
A094519_list = list(islice(A094519_gen(),20)) # Chai Wah Wu, Dec 26 2021

Formula

A094518(a(n)) > 0.