login
A045619
Numbers that are the products of 2 or more consecutive integers.
17
0, 2, 6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 110, 120, 132, 156, 182, 210, 240, 272, 306, 336, 342, 360, 380, 420, 462, 504, 506, 552, 600, 650, 702, 720, 756, 812, 840, 870, 930, 990, 992, 1056, 1122, 1190, 1260, 1320, 1332, 1406, 1482, 1560, 1640, 1680
OFFSET
1,2
COMMENTS
Erdős and Selfridge proved that, apart from the first term, these are never perfect powers (A001597). - T. D. Noe, Oct 13 2002
Numbers of the form x!/y! with y+1 < x. - Reinhard Zumkeller, Feb 20 2008
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T.D. Noe)
P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois Jour. Math. 19 (1975), 292-301.
FORMULA
a(n) = A000142(A137911(n))/A000142(A137912(n)-1) for n>1. - Reinhard Zumkeller, Feb 27 2008
Since the oblong numbers (A002378) have relative density of 100%, we have a(n) ~ (n-1) n ~ n^2. - Daniel Forgues, Mar 26 2012
a(n) = n^2 - 2*n^(5/3) + O(n^(4/3)). - Charles R Greathouse IV, Aug 27 2013
EXAMPLE
30 is in the sequence as 30 = 5*6 = 5*(5+1). - David A. Corneth, Oct 19 2021
MATHEMATICA
maxNum = 1700; lst = {}; For[i = 1, i <= Sqrt[maxNum], i++, j = i + 1; prod = i*j; While[prod < maxNum, AppendTo[lst, prod]; j++; prod *= j]]; lst = Union[lst]
PROG
(Python)
import heapq
from sympy import sieve
def aupton(terms, verbose=False):
p = 6; h = [(p, 2, 3)]; nextcount = 4; aset = {0, 2}
while len(aset) < terms:
(v, s, l) = heapq.heappop(h)
aset.add(v)
if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} i]")
if v >= p:
p *= nextcount
heapq.heappush(h, (p, 2, nextcount))
nextcount += 1
v //= s; s += 1; l += 1; v *= l
heapq.heappush(h, (v, s, l))
return sorted(aset)
print(aupton(52)) # Michael S. Branicky, Oct 19 2021
(PARI) list(lim)=my(v=List([0]), P, k=1, t); while(1, k++; P=binomial('n+k-1, k)*k!; if(subst(P, 'n, 1)>lim, break); for(n=1, lim, t=eval(P); if(t>lim, next(2)); listput(v, t))); Set(v) \\ Charles R Greathouse IV, Nov 16 2021
KEYWORD
easy,nonn,nice
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2000
More terms from Reinhard Zumkeller, Feb 27 2008
Incorrect program removed by David A. Corneth, Oct 19 2021
STATUS
approved