A162651 Numbers which can be expressed as the product of 3 positive integers in arithmetic progression.

1, 6, 8, 15, 24, 27, 28, 45, 48, 60, 64, 66, 80, 91, 105, 120, 125, 153, 162, 168, 190, 192, 210, 216, 224, 231, 276, 280, 288, 312, 315, 325, 336, 343, 360, 378, 384, 405, 435, 440, 480, 496, 504, 510, 512, 528, 561, 585, 624, 627, 630, 640, 648, 693, 703, 720

Offset: 1

Author

Franklin T. Adams-Watters, Jul 08 2009

Keywords

• nonn

Comments

Numbers of the form i*(i+j)*(i+2j), where i > 0 and j >= 0.

Examples

			1 = 1*1*1, 6 = 1*2*3, 8 = 2*2*2, 15 = 1*3*5, 24 = 2*3*4.
120 = 1*8*15 = 2*6*10 = 4*5*6.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000578, A007531, A000384, A033996, A011199.

Programs

Maple

N:= 1000: # for all terms <= N
S:= {}:
for i from 1 to floor(N^(1/3)) do
  S:= S union {seq(i*(i+j)*(i+2*j),j=0..floor((sqrt(i^4 + 8*i*N)-3*i^2)/(4*i)))}
od:
A:= sort(convert(S,list)); # Robert Israel, Feb 05 2020

PARI

al(n)={local(v,inc,prd);
v=vector(n);inc=[0];prd=[1];
for(k=1,n,
v[k]=vecmin(prd);
if(v[k]==prd[ #prd],inc=concat(inc,[0]);prd=concat(prd,[(#inc)^3]));
for(j=1,#prd,if(prd[j]==v[k],inc[j]++;prd[j]=j*(j+inc[j])*(j+2*inc[j]))));
v}

Python

from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A162651_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        for r in divisors(m,generator=True):
            if is_square(r**2-m//r):
                yield m
                break
A162651_list = list(islice(A162651_gen(),20)) # Chai Wah Wu, Jul 03 2023