cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A334174 Numbers that can be written as a product of two or more consecutive factorial numbers.

Original entry on oeis.org

1, 2, 12, 144, 288, 2880, 17280, 34560, 86400, 2073600, 3628800, 12441600, 24883200, 203212800, 435456000, 10450944000, 14631321600, 62705664000, 125411328000, 146313216000, 1316818944000, 17557585920000, 73741860864000, 144850083840000, 421382062080000
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 17 2020

Keywords

Examples

			    1 = 0! * 1!;
    2 = 1! * 2!;
   12 = 2! * 3!;
  144 = 3! * 4!;
  288 = 2! * 3! * 4!.

Links

Crossrefs

Cf. A000142, A000178, A001013, A010790, A034882, A045619, A058295, A334175.

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

Views

Author

Rémy Sigrist, Oct 17 2021

Keywords

Comments

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

Links

Crossrefs

Cf. A045619, A346928.

Programs

  • PARI
    See Links section.
  • Python
    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

Formula

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

A084720 Primes p such that p+1 is a product of two or more consecutive integers.

Original entry on oeis.org

5, 11, 19, 23, 29, 41, 59, 71, 89, 109, 131, 181, 239, 271, 359, 379, 419, 461, 503, 599, 701, 719, 811, 839, 929, 991, 1259, 1319, 1481, 1559, 1721, 1979, 2069, 2161, 2351, 2549, 2729, 2861, 2969, 3023, 3079, 3191, 3359, 3539, 3659, 4079, 4159, 4289, 4421
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth ((menakan_sAT)yahoo.com), Jun 11 2003

Keywords

Examples

			59 is a member as 60 = 3*4*5, 89 is a member as 90 = 9*10.

Crossrefs

Cf. A045619.

Extensions

More terms from David Wasserman, Jan 03 2005

A345708 a(n) is the least positive number starting an interval of consecutive integers whose product of elements is n.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Jun 24 2021

Keywords

Comments

This sequence is similar to A118235; here we multiply, there we add.
a(n) is the index of the first row of A068424 (interpreted as a square array) containing n.
If n is the product of k consecutive integers, then k! divides n.

Examples

			The square array A068424(n, k) begins:
  n\k|   1    2     3      4       5        6
  ---+---------------------------------------
    1|   1    2     6     24     120      720
    2|   2    6    24    120     720     5040
    3|   3   12    60    360    2520    20160
    4|   4   20   120    840    6720    60480
- so a(1) = a(2) = a(6) = a(24) = a(120) = a(720) = 1,
     a(3) = a(12) = a(60) = a(360) = 3,
     a(4) = a(20) = 4.

Links

Crossrefs

Cf. A000142, A001710, A045619, A068424, A118235, A246655.

Programs

  • PARI
    a(n) = { fordiv (n, d, my (r=n); for (k=d, oo, if (r==1, return (d), r%k, break, r/=k))) }
  • PARI
    a(n) = { for (i=2, oo, if (n%i!, forstep (j=i-1, 2, -1, my (d=sqrtnint(n,j)); while (d-1 && n%(d-1)==0, d--); while (n%d==0, my (r=n); for
(k=d, oo, if (r==1, return (if (d==2, 1, d)), r%k, break, r/=k)); d++)); break)); return (n) }
  • Python
    from sympy import divisors
def a(n):
    if n%2 == 0: return n
    divs = divisors(n)
    for i, d in enumerate(divs[:len(divs)//2]):
        p = lastj = d
        for j in divs[i+1:]:
            if p >= n or j - lastj > 1: break
            p, lastj = p*j, j
        if p == n: return d
    return n
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, Jun 29 2021

Formula

a(n) = 1 iff n is a factorial number (A000142).
a(n) <> 2.
a(n) = 3 iff n >= 3 and n belongs to A001710.
a(n) <= n.
a(p! / (n-1)!) = n for any n >= 3 and any prime number p >= n.
a(q) = q for any prime power q > 2.
a(n) = n for any odd number n.
a(n) < n iff n belongs to A045619.

A274187 Least number that is the product of n consecutive positive numbers and the product of 2 oblong numbers.

Original entry on oeis.org

4, 12, 24, 24, 120, 5040, 5040, 362880, 362880, 3628800, 39916800, 6227020800, 6227020800, 3379030566912000
Offset: 1

Views

Author

Gionata Neri, Jun 12 2016

Keywords

Examples

			a(3) = 24 = 2*3*4 = 2*12.
a(6) = 5040 = 2*3*4*5*6*7 = 12*420 = 56*90.

Crossrefs

Cf. A045619, A072389.

Programs

  • Maple
    N:= 10^10: # to get all terms <= N
A072389:= {seq(seq(n*(n+1)*m*(m+1),m=n..floor((sqrt(1+4*N/(n*(n+1))-1)/2))),n=1..floor((sqrt(1+2*N)-1)/2))}:
for n from 1 do
  x:= n!;
  for m from 1 while x <= N and not member(x, A072389) do
    x:= x*(n+m)/m
  od;
  if x > N then break fi;
  A[n]:= x;
od:
seq(A[i],i=1..n-1); # Robert Israel, Jun 16 2016

Extensions

a(14) from Robert Israel, Jun 16 2016

A304579 a(n) = (n^2 + 1)*(n^2 + 2).

Original entry on oeis.org

2, 6, 30, 110, 306, 702, 1406, 2550, 4290, 6806, 10302, 15006, 21170, 29070, 39006, 51302, 66306, 84390, 105950, 131406, 161202, 195806, 235710, 281430, 333506, 392502, 459006, 533630, 617010, 709806, 812702, 926406, 1051650, 1189190, 1339806, 1504302, 1683506
Offset: 0

Views

Author

Vincenzo Librandi, May 17 2018

Keywords

Comments

a(n) and A304578(n) are coprime for all n.

Links

Crossrefs

Cf. A002522, A304578.
Subsequence of A002378, A045619, A279019.

Programs

  • Magma
    [(n^2+1)*(n^2+2): n in [0..40]];
  • Mathematica
    CoefficientList[Series[2 (1 - 2 x + 10 x^2 + 3 x^4) / (1 - x)^5, {x, 0, 35}], x] (* or *) Table[(n^2 + 1) (n^2 + 2), {n, 0, 40}]
LinearRecurrence[{5,-10,10,-5,1},{2,6,30,110,306},40] (* Harvey P. Dale, Nov 13 2022 *)
  • PARI
    a(n) = my(k=n^2+1); k*(k+1); \\ Altug Alkan, May 17 2018

Formula

G.f.: 2*(1 - 2*x + 10*x^2 + 3*x^4)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A002378(A002522(n)). - Altug Alkan, May 17 2018
Sum_{n>=0} 1/a(n) = 1/4 + coth(Pi)*Pi/2 - coth(sqrt(2)*Pi)*Pi/(2*sqrt(2)). - Amiram Eldar, Feb 24 2023

A325480 a(n) is the largest integer m such that the product of n consecutive integers starting at m is divisible by at most n primes.

Original entry on oeis.org

16, 24, 24, 45, 48, 49, 120, 120, 125, 189, 240, 240, 350, 350, 350, 350, 374, 494, 494, 714, 714, 714, 714, 825, 832, 1078, 1078, 1078, 1078, 1425, 1440, 1440, 1856, 2175, 2175, 2175, 2175, 2175, 2175, 2175, 2870, 2870, 2870, 2871, 2880, 2880, 2880, 3219
Offset: 3

Views

Author

Onno M. Cain, Sep 06 2019

Keywords

Comments

Each term is only conjectured and has been verified up to 10^6.
Note a(2) is undefined if there are infinitely many Mersenne primes.

Examples

			For example, a(3) = 16 because 16 * 17 * 18 = 2^5 * 3^2 * 17 admits only three prime divisors (2, 3, and 17) and appears to be the largest product of three consecutive integers with the property.

Crossrefs

Cf. A002182, A045619, A163264, A164799, A239035, A244656, A006549.

Programs

  • SageMath
    for r in range(3, 100):
  history = []
  M = 0
  for n in range(1, 100000):
    primes = {p for p, _ in factor(n)}
    history.append(primes)
    history = history[-r:]
    total = set()
    for s in history: total |= s
    # Skip if too many primes.
    if len(total) > r: continue
    if n > M: M = n
  print(r, M-r+1)
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.