A045619 -id:A045619 - cp's OEIS frontend

A137912 Greatest of at least two consecutive numbers whose product is A045619(n).

1, 2, 3, 4, 5, 4, 6, 7, 8, 5, 9, 10, 11, 5, 12, 13, 14, 7, 16, 17, 18, 8, 19, 6, 20, 21, 22, 9, 23, 24, 25, 26, 27, 6, 28, 29, 7, 30, 31, 11, 32, 33, 34, 35, 36, 12, 37, 38, 39, 40, 41, 8, 13, 42, 43, 44, 45, 46, 47, 14, 48, 49, 50, 7, 51, 52, 15, 53, 54, 55, 9, 56, 57, 58, 16, 59, 60

Offset: 1

Reinhard Zumkeller, Feb 27 2008

nonn

A045619(n) = A000142(A137911(n))/A000142(a(n)-1) for n>1.

R. Zumkeller, Table of n, a(n) for n = 1..5000

Cf. A000142, A045619, A137911.

A137911 Smallest of at least two consecutive numbers whose product is A045619(n).

Original entry on oeis.org

0, 1, 1, 3, 4, 1, 5, 6, 7, 3, 8, 9, 10, 1, 11, 12, 13, 5, 15, 16, 17, 6, 18, 3, 19, 20, 21, 7, 22, 23, 24, 25, 26, 1, 27, 28, 4, 29, 30, 9, 31, 32, 33, 34, 35, 10, 36, 37, 38, 39, 40, 5, 11, 41, 42, 43, 44, 45, 46, 12, 47, 48, 49, 3, 50, 51, 13, 52, 53, 54, 6, 55, 56, 57, 14, 58, 59, 60

Offset: 1

Reinhard Zumkeller, Feb 27 2008

nonn

A045619(n)=A000142(a(n))/A000142(A137912(n)-1) for n>1.

			a(24) = 3, A137912(24) = 6: A045619 = 3*4*5*6 = 360.

R. Zumkeller, Table of n, a(n) for n = 1..5000

A053625 Product of 6 consecutive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 5040, 20160, 60480, 151200, 332640, 665280, 1235520, 2162160, 3603600, 5765760, 8910720, 13366080, 19535040, 27907200, 39070080, 53721360, 72681840, 96909120, 127512000, 165765600, 213127200, 271252800, 342014400, 427518000, 530122320

Offset: 0

Henry Bottomley, Mar 20 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002378, A007531, A045619, A052762, A052787.

Cf. A000579, A173333.

F:=Factorial;; Concatenation([0,0,0,0,0,0], List([6..30], n-> F(n)/F(n-5) )); # G. C. Greubel, Aug 27 2019

I:=[0,0,0,0,0,0,720]; [n le 7 select I[n] else 7*Self(n-1) -21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6) +Self(n-7): n in [1..30]]; // Vincenzo Librandi, Apr 28 2012

seq(combinat[numbperm](n, 6), n=0..31); # Zerinvary Lajos, Apr 26 2007

CoefficientList[Series[720*x^6/(1-x)^7,{x,0,30}],x] (* Vincenzo Librandi, Apr 28 2012 *)
Times@@@Partition[Range[-5,30],6,1] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,0,0,720},30] (* Harvey P. Dale, Nov 13 2015 *)
Pochhammer[Range[30]-6, 6] (* G. C. Greubel, Aug 27 2019 *)

a(n)=factorback([n-5..n]) \\ Charles R Greathouse IV, Oct 07 2015

[rising_factorial(n-5,6) for n in (0..30)] # G. C. Greubel, Aug 27 2019

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5) = n!/(n-6)! = A052787(n)*(n-6) = a(n-1)*n/(n-6).
E.g.f.: x^6*exp(x).
a(n) = 720 * A000579(n). - Zerinvary Lajos, Apr 26 2007
For n > 5: a(n) = A173333(n, n-6). - Reinhard Zumkeller, Feb 19 2010
G.f.: 720*x^6/(1-x)^7. - Colin Barker, Mar 27 2012
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Vincenzo Librandi, Apr 28 2012
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=6} 1/a(n) = 1/600.
Sum_{n>=6} (-1)^n/a(n) = 4*log(2)/15 - 661/3600. (End)

A064224 Numbers having more than one representation as the product of consecutive integers > 1.

Original entry on oeis.org

120, 210, 720, 5040, 175560, 17297280, 19958400, 259459200, 20274183401472000, 25852016738884976640000, 368406749739154248105984000000

Offset: 1

Jon Perry, Sep 22 2001

Solutions to the equations: x(x+1)...(x+m)=y(y+1)...(y+n) with x>1, y>1.
Some patterns are impossible, e.g. x(x+1)(x+2)(x+3)=y(y+1) has been proved impossible.
The early terms in this sequence each have two representations. Is two the maximum possible? The sequence is infinite: for any n, the number n*(n+1)*...*(n^2+n-1) is in this sequence. The next number of this form is 20274183401472000, which is obtained when n=4. - T. D. Noe, Nov 22 2004
Using an improved algorithm I have performed an exhaustive search up to 2.15 * 10^33 and can confirm the terms shown above are all that exist up to that point. For all k = A002378(n) > 2 we can construct a member of this sequence by equating n(n+1)(n+2)...(k-1) to (n+2)(n+3)...(k-1)k. Also, as demonstrated in my examples below, 5040 is related to 720 as 259459200 is to 210. So we also know that 36055954861352887137197787308347629783163600896000000000 and 6244042313569035223343873483125151604764341428027427022254596874567680000000000000 are terms. - Robert Munafo, Aug 17 2007 [edited by Peter Munn, Aug 20 2023]
MacLeod and Barrodale prove that the equation x(x+1)...(x+m-1) = y(y+1)...(y+n-1) has no solutions x>1 and y>1 for the following pairs of (m,n): (2,4), (2,6), (2,8), (2,12), (4,8), (5,10). They also show that (2,3) has two solutions and (3,6) has one solution. They conjecture that (2,k) has no solution for k>3. [T. D. Noe, Jul 29 2009]

			120 is here because 120 = 2*3*4*5 = 4*5*6.
a(2)=210 because we can write 210=5*6*7 or 14*15. The term a(8) = 259459200 = 5*6*7*8*9*10*11*12*13 = 8*9*10*11*12*13*14*15 is related to 210 by adding the intervening integers (8 through 13) to both products.

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.
R. A. MacLeod and I. Barrodale, On equal products of consecutive integers, Canad. Math. Bull., 13 (1970) 255-259. [_T. D. Noe_, Jul 29 2009]
Robert Munafo, Page dealing with this sequence

Cf. A003015 (numbers occurring 5 or more times in Pascal's triangle).
Cf. A002378, A160642.
Subsequence of A045619, A100934.
Cf. A163263 (non-overlapping case). [T. D. Noe, Jul 29 2009]

nn=10^10; t3={}; Do[m=0; p=n; While[m++; p=p(n+m); p<=nn, t3={t3, p}], {n, 2, 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 = 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 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(8, verbose=True)) # Michael S. Branicky, Jun 24 2021

a(1), a(7) and a(8) from T. D. Noe, Nov 22 2004
a(9) and a(10) from Robert Munafo, Aug 13 2007
a(11) from Robert Munafo, Aug 17 2007
Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar

A137895 Products of two or more consecutive numbers that do not have prime gaps in their factorizations.

Original entry on oeis.org

2, 6, 12, 24, 30, 60, 72, 90, 120, 210, 240, 360, 420, 600, 720, 840, 1260, 1680, 2520, 3360, 5040, 6480, 6720, 9240, 15120, 15750, 20160, 30240, 40320, 50400, 55440, 60480, 117600, 147840, 151200, 166320, 181440, 194040, 240240, 291060, 332640

Offset: 1

Reinhard Zumkeller, Feb 20 2008

nonn

Intersection of A045619 and A073491; A000142 is a subsequence apart from initial terms.

			a(10) = 210 = 2 * 3 * 5 * 7 = 5 * (5+1) * (5+2);
a(11) = 240 = 2^4 * 3 * 5 = 15 * (15+1);
a(12) = 360 = 2^3 * 3^2 * 5 = 3 * (3+1) * (3+2) * (3+3);
a(13) = 420 = 2^2 * 3 * 5 * 7 = 20 * (20+1);
a(14) = 600 = 2^3 * 3 * 5^2 = 24 * (24+1);
a(15) = 720 = 2^4 * 3^2 * 5 = 6!.

David A. Corneth, Table of n, a(n) for n = 1..193

Subsequence of A055932.

Cf. A045619, A073491, A000142.

A137899 Number of partitions of n into products of at least two consecutive positive integers.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 0, 4, 0, 6, 0, 7, 0, 7, 0, 11, 0, 12, 0, 12, 0, 17, 0, 19, 0, 19, 0, 26, 0, 28, 0, 29, 0, 38, 0, 42, 0, 43, 0, 55, 0, 60, 0, 62, 0, 77, 0, 85, 0, 87, 0, 108, 0, 118, 0, 122, 0, 147, 0, 161, 0, 166, 0, 200, 0, 217, 0, 225, 0, 264, 0, 289, 0, 299, 0, 351

Offset: 1

Reinhard Zumkeller, Feb 21 2008

nonn

a(odd) = 0.

			a(12)=#{3*4,2*3+2*3,2*3+1*2+1*2+1*2,1*2+1*2+1*2+1*2+1*2+1*2}=4.

R. Zumkeller, Table of n, a(n) for n = 1..250

Cf. A045619, A137900, A000041.

A137900 Number of partitions of 2*n into distinct products of at least two consecutive positive integers.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 3, 4, 1, 2, 4, 2, 2, 5, 3, 3, 7, 4, 3, 6, 3, 5, 8, 4, 5, 8, 5, 5, 9, 6, 6, 12, 8, 6, 12, 8, 8, 13, 8, 8, 14, 10, 9, 16, 12, 10, 18, 13, 9, 19, 15, 14, 23, 16, 13, 22, 17, 14, 24, 20, 16, 29, 24, 16, 29, 24, 21, 33, 26, 21, 33, 29, 24

Offset: 1

Reinhard Zumkeller, Feb 21 2008

nonn

There are no partitions of odd numbers into products of two or

more consecutive numbers: A137899(odd) = 0.

			a(16) = #{5*6+1*2, 2*3*4+2*3+1*2, 4*5+3*4} = 3.

R. Zumkeller, Table of n, a(n) for n = 1..250

Cf. A045619, A137899, A000009.

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.

A227027 Triangular numbers representable as x!/y! with y < x-1.

Original entry on oeis.org

6, 120, 210, 990, 7140, 185136, 242556, 2162160, 8239770, 258474216, 279909630, 9508687656, 323015470680, 10973017315470, 372759573255306, 12662852473364940, 430164224521152660, 14612920781245825506, 496409142337836914550

Offset: 1

Alex Ratushnyak, Jun 27 2013

nonn

Triangular numbers in A045619, except A045619(1)=0. The sequence is infinite because A029549 is a subsequence. According to Melissen's comment in A097571,  y > x-7.
The sequence of x's producing a(n): A227026.
a(2) and a(3) have two representations:
a(2) = 120 = 5*4*3*2 = 6*5*4.
a(3) = 210 = 7*6*5 = 15*14.

			990 is in the sequence since 990 = 11!/8! = 11*10*9 is a ratio of factorials and 990 = (44)(44 + 1)/2 is a triangular number.

Cf. A000217, A001219, A029549, A045619, A097571, A227026.

A093449 Least number with n distinct prime divisors arising as the product of two or more consecutive integers.

Original entry on oeis.org

2, 6, 30, 210, 2730, 39270, 510510, 23393370, 363993630, 64790866140, 530514844860, 126408523110870, 3425113062060690, 660393717163700520, 26657280574571657010, 3448055881024876471350, 308480161111936386482910

Offset: 1

Amarnath Murthy, Apr 03 2004

2, 6, 30, 210 and 510510 are primorials (A002110). There are no more primorials in the first 300 terms.

Upper bounds for a(14)-a(18): 660393717163700520, 28386773771493397260, 3448055881024876471350, 308480161111936386482910, 32521466098360753728404190.

			a(7) = 510510 = 714*715 has prime divisors 2, 3, 5, 7, 11, 13 and 17.

Cf. A002110, A045619, A093450.

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

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

cp's OEIS Frontend

A137912 Greatest of at least two consecutive numbers whose product is A045619(n).

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A137911 Smallest of at least two consecutive numbers whose product is A045619(n).

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A053625 Product of 6 consecutive integers.

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A064224 Numbers having more than one representation as the product of consecutive integers > 1.

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A137895 Products of two or more consecutive numbers that do not have prime gaps in their factorizations.

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A137899 Number of partitions of n into products of at least two consecutive positive integers.

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A137900 Number of partitions of 2*n into distinct products of at least two consecutive positive integers.

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

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A227027 Triangular numbers representable as x!/y! with y < x-1.

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A093449 Least number with n distinct prime divisors arising as the product of two or more consecutive integers.