A100934 Numbers having more than one representation as the product of consecutive integers.

6, 24, 120, 210, 720, 5040, 40320, 175560, 362880, 3628800, 17297280, 19958400, 39916800, 259459200, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 20274183401472000, 121645100408832000

Offset: 1

Author

T. D. Noe, Nov 22 2004

Keywords

• nonn

Comments

All the factorials occur because we allow products to start with 1. See A064224 for a more restrictive case.

Examples

			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.
		

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.

Crossrefs

Cf. A064224, A003015 (numbers occurring 5 or more times in Pascal's triangle).

Programs

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]

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

Extensions

a(18) and beyond from Michael S. Branicky, Jun 24 2021