login
A162651
Numbers which can be expressed as the product of 3 positive integers in arithmetic progression.
2
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
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
KEYWORD
nonn
AUTHOR
STATUS
approved