A093449 -id:A093449

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A093450

Number of consecutive integers whose product = A093449(n).

+20
3

2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..17.

EXAMPLE

a(5) = 3 because A093449(5) = 13*14*15.

CROSSREFS

Cf. A093449.

KEYWORD

hard,more,nonn

AUTHOR

Amarnath Murthy, Apr 03 2004

EXTENSIONS

Edited, corrected and extended by David Wasserman, Mar 21 2007

a(14)-a(17) from Donovan Johnson, Sep 13 2008

STATUS

approved

A045619

Numbers that are the products of 2 or more consecutive integers.

+10
17

0, 2, 6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 110, 120, 132, 156, 182, 210, 240, 272, 306, 336, 342, 360, 380, 420, 462, 504, 506, 552, 600, 650, 702, 720, 756, 812, 840, 870, 930, 990, 992, 1056, 1122, 1190, 1260, 1320, 1332, 1406, 1482, 1560, 1640, 1680

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Erdős and Selfridge proved that, apart from the first term, these are never perfect powers (A001597). - T. D. Noe, Oct 13 2002

Numbers of the form x!/y! with y+1 < x. - Reinhard Zumkeller, Feb 20 2008

LINKS

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T.D. Noe)

P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois Jour. Math. 19 (1975), 292-301.

FORMULA

a(n) = A000142(A137911(n))/A000142(A137912(n)-1) for n>1. - Reinhard Zumkeller, Feb 27 2008

Since the oblong numbers (A002378) have relative density of 100%, we have a(n) ~ (n-1) n ~ n^2. - Daniel Forgues, Mar 26 2012

a(n) = n^2 - 2*n^(5/3) + O(n^(4/3)). - Charles R Greathouse IV, Aug 27 2013

EXAMPLE

30 is in the sequence as 30 = 5*6 = 5*(5+1). - David A. Corneth, Oct 19 2021

MATHEMATICA

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]

PROG

(Python)

import heapq

from sympy import sieve

def aupton(terms, verbose=False):

p = 6; h = [(p, 2, 3)]; nextcount = 4; aset = {0, 2}

while len(aset) < terms:

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

aset.add(v)

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

if v >= p:

p *= nextcount

heapq.heappush(h, (p, 2, nextcount))

nextcount += 1

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

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

return sorted(aset)

print(aupton(52)) # Michael S. Branicky, Oct 19 2021

(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

CROSSREFS

Union of A002378, A007531, A052762, A052787, A053625, etc. - R. J. Mathar, Oct 19 2021

Cf. A001597, A000142, A137895, A093449, A064224, A084720, A137899, A137900, A120436, A097889.

KEYWORD

easy,nonn,nice

AUTHOR

Erich Friedman

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2000

More terms from Reinhard Zumkeller, Feb 27 2008

Incorrect program removed by David A. Corneth, Oct 19 2021

STATUS

approved

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