A045619 - OEIS (original) (raw)

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

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

FORMULA

Since the oblong numbers (A002378) have relative density of 100%, we have a(n) ~ (n-1) n ~ n^2. - Daniel Forgues, Mar 26 2012

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)

(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

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2000