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

1,1

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

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

Sequence in context: A217193 A109583 A352807 * A127917 A293118 A293121

Adjacent sequences: A100931 A100932 A100933 * A100935 A100936 A100937

Keyword

nonn

Author

T. D. Noe, Nov 22 2004

Extensions

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

Status

approved