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%I #50 Nov 16 2021 21:55:15
%S 0,2,6,12,20,24,30,42,56,60,72,90,110,120,132,156,182,210,240,272,306,
%T 336,342,360,380,420,462,504,506,552,600,650,702,720,756,812,840,870,
%U 930,990,992,1056,1122,1190,1260,1320,1332,1406,1482,1560,1640,1680
%N Numbers that are the products of 2 or more consecutive integers.
%C Erdős and Selfridge proved that, apart from the first term, these are never perfect powers (A001597). - _T. D. Noe_, Oct 13 2002
%C Numbers of the form x!/y! with y+1 < x. - _Reinhard Zumkeller_, Feb 20 2008
%H Michael S. Branicky, <a href="/A045619/b045619.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T.D. Noe)
%H P. Erdős and J. L. Selfridge, <a href="http://projecteuclid.org/euclid.ijm/1256050816">The product of consecutive integers is never a power</a>, Illinois Jour. Math. 19 (1975), 292-301.
%F a(n) = A000142(A137911(n))/A000142(A137912(n)-1) for n>1. - _Reinhard Zumkeller_, Feb 27 2008
%F Since the oblong numbers (A002378) have relative density of 100%, we have a(n) ~ (n-1) n ~ n^2. - _Daniel Forgues_, Mar 26 2012
%F a(n) = n^2 - 2*n^(5/3) + O(n^(4/3)). - _Charles R Greathouse IV_, Aug 27 2013
%e 30 is in the sequence as 30 = 5*6 = 5*(5+1). - _David A. Corneth_, Oct 19 2021
%t 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]
%o (Python)
%o import heapq
%o from sympy import sieve
%o def aupton(terms, verbose=False):
%o p = 6; h = [(p, 2, 3)]; nextcount = 4; aset = {0, 2}
%o while len(aset) < terms:
%o (v, s, l) = heapq.heappop(h)
%o aset.add(v)
%o if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} i]")
%o if v >= p:
%o p *= nextcount
%o heapq.heappush(h, (p, 2, nextcount))
%o nextcount += 1
%o v //= s; s += 1; l += 1; v *= l
%o heapq.heappush(h, (v, s, l))
%o return sorted(aset)
%o print(aupton(52)) # _Michael S. Branicky_, Oct 19 2021
%o (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
%Y Union of A002378, A007531, A052762, A052787, A053625, etc. - _R. J. Mathar_, Oct 19 2021
%Y Cf. A001597, A000142, A137895, A093449, A064224, A084720, A137899, A137900, A120436, A097889.
%K easy,nonn,nice
%O 1,2
%A _Erich Friedman_
%E More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2000
%E More terms from _Reinhard Zumkeller_, Feb 27 2008
%E Incorrect program removed by _David A. Corneth_, Oct 19 2021
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