A064224 -id:A064224 - cp's OEIS frontend

A045619 Numbers that are the products of 2 or more consecutive integers.

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

Offset: 1

Erich Friedman

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

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

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.

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.

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]

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

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

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

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

A100934 Numbers having more than one representation as the product of consecutive integers.

Original entry on oeis.org

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

T. D. Noe, Nov 22 2004

nonn

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

			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.

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.

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

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]

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

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

A163264 Highly composite numbers that are the product of consecutive integers.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Jul 28 2009

nonn

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.

			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.

Cf. A002182, A045619, A064224.

A163263 Numbers having multiple representations as the product of non-overlapping ranges of consecutive numbers.

Original entry on oeis.org

210, 720, 175560, 17297280

Offset: 1

T. D. Noe, Jul 29 2009

nonn

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.

			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.

Art Kalb, Why Number Theory is Hard, YouTube video, 2024
Carlos Rivera, Puzzle 469. 5040, The Prime Puzzles and Problems Connection.

Cf. A064224.

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

A175340 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.

Original entry on oeis.org

1680, 4320, 120960, 166320, 175560, 215760, 725760, 1080647568000

Offset: 1

Manuel Valdivia, Apr 16 2010, Apr 18 2010

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)

			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)

Cf. A002808, A064224, A163263, A342876.

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.

Definition edited by N. J. A. Sloane, Apr 18 2010

Keyword:base removed by R. J. Mathar, Apr 24 2010

A360662 Numbers having more than one representation as the product of at least two consecutive odd integers > 1.

Original entry on oeis.org

135135, 2110886623587616875, 118810132577324221759073444371080321140625, 262182986027006205192949807157375529898104505103011391412633845449072265625

Offset: 1

Ilya Gutkovskiy, Feb 15 2023

nonn

The next term is too large to include.

			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.

Cf. A001147, A064224, A100934, A257836, A343459.

a(4) from Alois P. Heinz, Feb 28 2023

cp's OEIS Frontend

A045619 Numbers that are the products of 2 or more consecutive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A100934 Numbers having more than one representation as the product of consecutive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A163264 Highly composite numbers that are the product of consecutive integers.

Views

Author

Keywords

Comments

Examples

Crossrefs

A163263 Numbers having multiple representations as the product of non-overlapping ranges of consecutive numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

A175340 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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A360662 Numbers having more than one representation as the product of at least two consecutive odd integers > 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions