A100934 - OEIS

A100934

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

%I #19 Jun 28 2021 02:48:43

%S 6,24,120,210,720,5040,40320,175560,362880,3628800,17297280,19958400,

%T 39916800,259459200,479001600,6227020800,87178291200,1307674368000,

%U 20922789888000,355687428096000,6402373705728000,20274183401472000,121645100408832000

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

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

%H Michael S. Branicky, <a href="/A100934/b100934.txt">Table of n, a(n) for n = 1..26</a>

%H H. L. Abbott, P. Erdos and D. Hanson, <a href="http://www.jstor.org/stable/2319526">On the numbers of times an integer occurs as a binomial coefficient</a>, Amer. Math. Monthly, (March 1974), 256-261.

%e 120 is a term since 120 = 1*2*3*4*5 = 2*3*4*5 = 4*5*6.

%e 210 is a term since 210 = 14*15 = 5*6*7.

%e Other non-factorial terms are:

%e 175560 = Product_{i=55..57} i = Product_{i=19..22} i,

%e 17297280 = Product_{i=63..66} i = Product_{i= 8..14} i,

%e 19958400 = Product_{i= 5..12} i = Product_{i= 3..11} i,

%e 259459200 = Product_{i= 8..15} i = Product_{i= 5..13} i,

%e 20274183401472000 = Product_{i=6..20} i = Product_{i=4..19} i.

%t 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]

%o (Python)

%o import heapq

%o def aupton(terms, verbose=False):

%o p = 1*2; h = [(p, 1, 2)]; nextcount = 3; alst = []; oldv = None

%o while len(alst) < terms:

%o (v, s, l) = heapq.heappop(h)

%o if v == oldv and v not in alst:

%o alst.append(v)

%o if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} i = Prod_{{i = {olds}..{oldl}}} i]")

%o if v >= p:

%o p *= nextcount

%o heapq.heappush(h, (p, 1, nextcount))

%o nextcount += 1

%o oldv, olds, oldl = v, s, l

%o v //= s; s += 1; l += 1; v *= l

%o heapq.heappush(h, (v, s, l))

%o return alst

%o print(aupton(20, verbose=True)) # _Michael S. Branicky_, Jun 24 2021

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

%K nonn

%O 1,1

%A _T. D. Noe_, Nov 22 2004

%E a(18) and beyond from _Michael S. Branicky_, Jun 24 2021