OFFSET
1,1
COMMENTS
All the factorials occur because we allow products to start with 1. See A064224 for a more restrictive case.
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..26
H. L. Abbott, P. Erdos and D. Hanson, On the numbers of times an integer occurs as a binomial coefficient, Amer. Math. Monthly, (March 1974), 256-261.
EXAMPLE
120 is a term since 120 = 1*2*3*4*5 = 2*3*4*5 = 4*5*6.
210 is a term since 210 = 14*15 = 5*6*7.
Other non-factorial terms are:
175560 = Product_{i=55..57} i = Product_{i=19..22} i,
17297280 = Product_{i=63..66} i = Product_{i= 8..14} i,
19958400 = Product_{i= 5..12} i = Product_{i= 3..11} i,
259459200 = Product_{i= 8..15} i = Product_{i= 5..13} i,
20274183401472000 = Product_{i=6..20} i = Product_{i=4..19} i.
MATHEMATICA
nn=10^10; t3={}; Do[m=0; p=n; While[m++; p=p(n+m); p<=nn, t3={t3, p}], {n, Sqrt[nn]}]; t3=Sort[Flatten[t3]]; lst={}; Do[If[t3[[i]]==t3[[i+1]], AppendTo[lst, t3[[i]]]], {i, Length[t3]-1}]; Union[lst]
PROG
(Python)
import heapq
def aupton(terms, verbose=False):
p = 1*2; h = [(p, 1, 2)]; nextcount = 3; alst = []; oldv = None
while len(alst) < terms:
(v, s, l) = heapq.heappop(h)
if v == oldv and v not in alst:
alst.append(v)
if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} i = Prod_{{i = {olds}..{oldl}}} i]")
if v >= p:
p *= nextcount
heapq.heappush(h, (p, 1, nextcount))
nextcount += 1
oldv, olds, oldl = v, s, l
v //= s; s += 1; l += 1; v *= l
heapq.heappush(h, (v, s, l))
return alst
print(aupton(20, verbose=True)) # Michael S. Branicky, Jun 24 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Nov 22 2004
EXTENSIONS
a(18) and beyond from Michael S. Branicky, Jun 24 2021
STATUS
approved