A137912 -id:A137912 - 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

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

A348401 a(n) is the least m > 0 such that n = m! / k! for some k <= m.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 4, 13, 14, 15, 16, 17, 18, 19, 5, 21, 22, 23, 4, 25, 26, 27, 28, 29, 6, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 7, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 8, 57, 58, 59, 5, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Rémy Sigrist, Oct 17 2021

nonn

For any n > 0, n appears in the a(n)-th row of A346928.

Rémy Sigrist, PARI program for A348401

Cf. A045619, A346928.

PARI
```
See Links section.
```

from math import factorial
def a(n):
    f = [factorial(i) for i in range(1, n+1)]
    for m, fm in enumerate(f, start=1):
        for fk in f[:m]:
            if n == fm // fk:
                return m
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, Oct 17 2021

a(n) <= n.
a(n) < n iff n belongs to A045619 \ {2}.
If n = A045619(k), then a(n) = A137912(k). - R. J. Mathar, Oct 19 2021

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

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

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A348401 a(n) is the least m > 0 such that n = m! / k! for some k <= m.

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A152739 Sqrt((even members of A118962)/2).

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