cp's OEIS Frontend

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

%I A162651 #8 Jul 03 2023 14:31:10
%S A162651 1,6,8,15,24,27,28,45,48,60,64,66,80,91,105,120,125,153,162,168,190,
%T A162651 192,210,216,224,231,276,280,288,312,315,325,336,343,360,378,384,405,
%U A162651 435,440,480,496,504,510,512,528,561,585,624,627,630,640,648,693,703,720
%N A162651 Numbers which can be expressed as the product of 3 positive integers in arithmetic progression.
%C A162651 Numbers of the form i*(i+j)*(i+2j), where i > 0 and j >= 0.
%H A162651 Robert Israel, <a href="/A162651/b162651.txt">Table of n, a(n) for n = 1..10000</a>
%e A162651 1 = 1*1*1, 6 = 1*2*3, 8 = 2*2*2, 15 = 1*3*5, 24 = 2*3*4.
%e A162651 120 = 1*8*15 = 2*6*10 = 4*5*6.
%p A162651 N:= 1000: # for all terms <= N
%p A162651 S:= {}:
%p A162651 for i from 1 to floor(N^(1/3)) do
%p A162651   S:= S union {seq(i*(i+j)*(i+2*j),j=0..floor((sqrt(i^4 + 8*i*N)-3*i^2)/(4*i)))}
%p A162651 od:
%p A162651 A:= sort(convert(S,list)); # _Robert Israel_, Feb 05 2020
%o A162651 (PARI) al(n)={local(v,inc,prd);
%o A162651 v=vector(n);inc=[0];prd=[1];
%o A162651 for(k=1,n,
%o A162651 v[k]=vecmin(prd);
%o A162651 if(v[k]==prd[ #prd],inc=concat(inc,[0]);prd=concat(prd,[(#inc)^3]));
%o A162651 for(j=1,#prd,if(prd[j]==v[k],inc[j]++;prd[j]=j*(j+inc[j])*(j+2*inc[j]))));
%o A162651 v}
%o A162651 (Python)
%o A162651 from itertools import count, islice
%o A162651 from sympy import divisors
%o A162651 from sympy.ntheory.primetest import is_square
%o A162651 def A162651_gen(startvalue=1): # generator of terms >= startvalue
%o A162651     for m in count(max(startvalue,1)):
%o A162651         for r in divisors(m,generator=True):
%o A162651             if is_square(r**2-m//r):
%o A162651                 yield m
%o A162651                 break
%o A162651 A162651_list = list(islice(A162651_gen(),20)) # _Chai Wah Wu_, Jul 03 2023
%Y A162651 Cf. A000578, A007531, A000384, A033996, A011199.
%K A162651 nonn
%O A162651 1,2
%A A162651 _Franklin T. Adams-Watters_, Jul 08 2009