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

1,2

Comments

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

Links

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

Example

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.

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

Prog

(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

Crossrefs

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

Sequence in context: A315926 A063534 A361420 * A275321 A022320 A318387

Adjacent sequences: A162648 A162649 A162650 * A162652 A162653 A162654

Keyword

nonn

Author

Franklin T. Adams-Watters, Jul 08 2009

Status

approved