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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
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
FORMULA
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
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
Numbers having more than one representation as the product of consecutive integers.
+10
3
6, 24, 120, 210, 720, 5040, 40320, 175560, 362880, 3628800, 17297280, 19958400, 39916800, 259459200, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 20274183401472000, 121645100408832000
COMMENTS
All the factorials occur because we allow products to start with 1. See A064224 for a more restrictive case.
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).
EXTENSIONS
a(18) and beyond from _Michael S. Branicky_, Jun 24 2021
Highly composite numbers that are the product of consecutive integers.
+10
3
2, 6, 12, 24, 60, 120, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 15120, 20160, 50400, 55440, 166320, 332640, 665280, 2162160, 3603600, 4324320, 8648640, 17297280, 32432400, 43243200
COMMENTS
Intersection of A002182 and A045619. Some of these numbers have two representations as the product of consecutive integers. The shortest representation is shown in the examples below. This sequence is probably complete.
EXAMPLE
2=1*2, 6=2*3, 12=3*4, 24=2*3*4, 60=3*4*5, 120=4*5*6, 240=15*16, 360=3*4*5*6, 720=8*9*10, 840=4*5*6*7, 1260=35*36, 1680=5*6*7*8, 2520=3*4*5*6*7, 5040=7*8*9*10, 15120=5*6*7*8*9, 20160=3*4*5*6*7*8, 50400=224*225, 55440=7*8*9*10*11, 166320=54*55*56, 332640=6*7*8*9*10*11, 665280=7*8*9*10*11*12, 2162160=9*10*11*12*13*14, 3603600=10*11*12*13*14*15, 4324320=2079*2080, 8648640=7*8*9*10*11*12*13, 17297280=63*64*65*66, 32432400=9*10*11*12*13*14*15, 43243200=350*351*352.
Numbers having multiple representations as the product of non-overlapping ranges of consecutive numbers.
+10
2
210, 720, 175560, 17297280
COMMENTS
A subsequence of A064224. This sequence gives solutions P to the equation P = (x+1)...(x+m) = (y+1)...(y+n) with x>0, y>0 and x+m < y+1. So far, no numbers P with more than two representations have been discovered. Note that the only the lowest range of consecutive numbers (x+1 to x+m) can contain prime numbers; the other ranges are in a gap between consecutive primes. Gaps between the first 45000 primes were searched for additional terms, but none were found.
EXAMPLE
210 = 5*6*7 = 14*15.
720 = 2*3*4*5*6 = 8*9*10.
175560 = 19*20*21*22 = 55*56*57.
17297280 = 8*9*10*11*12*13*14 = 63*64*65*66.
PROG
(Python)
import heapq
def aupton(terms, verbose=False):
p = 2*3; h = [(p, 2, 3)]; nextcount = 4; alst = []; oldv = None
while len(alst) < terms:
(v, s, l) = heapq.heappop(h)
if v == oldv and ((s > oldl) or (olds > l)) 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, 2, 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(4, verbose=True)) # _Michael S. Branicky_, Jun 24 2021
Numbers that can be written as a product of k consecutive composite numbers and also of k+1 consecutive composite numbers, for some k>1, with no factor used twice.
+10
2
1680, 4320, 120960, 166320, 175560, 215760, 725760, 1080647568000
COMMENTS
The term 725760 has three representations of 4, 5 and 6 numbers (with overlap).
From _David A. Corneth_, Mar 27 2021, Mar 28 2021: (Start)
a(12) > 10^22 if it exists.
Let x be the product of k consecutive composite numbers and y be the product of k+1 consecutive composite numbers giving some a(m). This sequence does not allow factors of x and y to overlap. If we do allow such overlaps we get A342876. (End)
FORMULA
Numbers of the form Product_{i=x..x+k} A002808(i) = Product_{i=y..y+k-1} A002808(i), where y > x + k and k > 1.
EXAMPLE
1680 = 10*12*14 = 40*42.
4320 = 6*8*9*10 = 15*16*18.
120960 = 8*9*10*12*14 = 16*18*20*21.
166320 = 18*20*21*22 = 54*55*56.
175560 = 55*56*57 = 418*420.
215760 = 58*60*62 = 464*465.
725760 = 12*14*15*16*18 = 27*28*30*32.
1080647568000 = 49*50*51*52*54*55*56 = 98*99*100*102*104*105.
From _David A. Corneth_, Mar 28 2021: (Start)
1814400 = 8*9*10*12*14*15 = 15*16*18*20*21 is not in the sequence as the factor 15 is used more than once.
104613949440000 = 12*14*15*16*18*20*21*22*24*25*26 = 20*21*22*24*25*26*27*28*30*32 is not here because, among others, the factor 20 is used more than once.
115880067072000 = 4*6*8*9*10*12*14*15*16*18*20*21*22 = 8*9*10*12*14*15*16*18*20*21*22*24 is not here because most factors are used more than once. (End)
AUTHOR
_Manuel Valdivia_, Apr 16 2010, Apr 18 2010
EXTENSIONS
Definition edited by _N. J. A. Sloane_, Apr 18 2010
Keyword:base removed by _R. J. Mathar_, Apr 24 2010
Numbers having more than one representation as the product of at least two consecutive odd integers > 1.
+10
0
135135, 2110886623587616875, 118810132577324221759073444371080321140625, 262182986027006205192949807157375529898104505103011391412633845449072265625
COMMENTS
The next term is too large to include.
EXAMPLE
135135 = 3 * 5 * 7 * 9 * 11 * 13 = 7 * 9 * 11 * 13 * 15.
2110886623587616875 = 5 * 7 * 9 * ... * 33 = 9 * 11 * 13 * ... * 35.
118810132577324221759073444371080321140625 = 7 * 9 * 11 * ... * 61 = 11 * 13 * 15 * ... * 63.
AUTHOR
_Ilya Gutkovskiy_, Feb 15 2023
EXTENSIONS
a(4) from _Alois P. Heinz_, Feb 28 2023
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