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 11-15 of 15 results.

A280714 Partial products of A055067.

Original entry on oeis.org

1, 1, 2, 6, 144, 2880, 2073600, 1306368000, 17557585920000, 637129677864960000, 2312016175036366848000000, 640890883720080890265600000000, 306987758727332698566646824960000000000, 136544225638605874463902854662848512000000000000
Offset: 1

Views

Author

Jaroslav Krizek, Jan 07 2017

Keywords

Comments

A055067(n) = the product of non-divisors of n.

Crossrefs

Cf. A000178, A055067, A092143, A280713.

Programs

  • Magma
    [&*[Factorial(k) / &*[d: d in Divisors(k)]: k in [1..n]]: n in [1..100]];
  • Mathematica
    FoldList[#1 #2 &, Table[Times @@ Complement[Range@ n, Divisors@ n], {n, 14}]] (* Michael De Vlieger, Jan 09 2017 *)

Formula

a(n) = Product_{i=1..n} A055067(i).
a(n) = A000178(n)/A092143(n). - Amiram Eldar, Aug 16 2025

A308820 a(n) = Product_{k=1..n} ceiling(n/k)!.

Original entry on oeis.org

1, 2, 12, 96, 2880, 34560, 5806080, 92897280, 25082265600, 2006581248000, 794606174208000, 19070548180992000, 208250386136432640000, 5831010811820113920000, 4198327784510482022400000, 3224315738504050193203200000, 14799609239733590386802688000000
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 26 2019

Keywords

Links

Crossrefs

Cf. A010786, A020696, A092143, A131385.

Programs

  • Magma
    [(&*[Factorial(Ceiling(n/(n-j+1))): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Mar 08 2023
  • Maple
    seq(mul(ceil(n/k)!, k=1..n), n=1..30); # Ridouane Oudra, Apr 10 2023
  • Mathematica
    a[n_] := Product[Ceiling[n/k]!, {k, 1, n}]; Table[a[n], {n, 1, 17}]
  • PARI
    a(n) = prod(k=1, n, ceil(n/k)!); \\ Michel Marcus, Jun 27 2019
  • SageMath
    def A308820(n): return product( factorial(ceil(n/(n-k+1))) for k in range(1,n+1))
[A308820(n) for n in range(1,21)] # G. C. Greubel, Mar 08 2023

Formula

a(n) = Product_{k=1..n-1} Product_{d|k} (d + 1).
a(n) = Product_{k=1..n-1} (k + 1)^floor((n-1)/k). - Ridouane Oudra, Apr 10 2023
a(n) = A131385(n)*A092143(n-1). - Ridouane Oudra, Sep 20 2024

A090494 Product_{j=1..n} Product_{k=1..n} lcm(j,k).

Original entry on oeis.org

1, 1, 8, 7776, 1146617856, 1289945088000000000, 46798828032806092800000000000, 2350577043461005964030008507760640000000000000, 8206262459636402163263383676462776103575725539328000000000000000, 2746781358330240881921653545637784861521126603512175621574459373964492800000000000000000
Offset: 0

Views

Author

N. J. A. Sloane, Feb 03 2004

Keywords

Crossrefs

Cf. A018806.
Cf. A092143, A092287, A129365, A129454.

Programs

  • Maple
    f := n->mul(mul(lcm(j,k),k=1..n),j=1..n);

Formula

Let p be a prime and let ordp(n,p) denote the exponent of the largest power of p which divides n. For example, ordp(48,2)=4 since 48 = 3*(2^4). Then the prime factorization of a(n) appears to be given by the formula ordp(a(n),p)= sum_{k >= 1} [(2*(p^k)-1)*floor((n/(p^k)))^2] + 2*sum_{k >= 1} [floor(n/(p^k))*mod(n,p^k)], for each prime p. See the comments sections of A092143, A092287, A129365 and A129454 for similar conjectural prime factorizations. - Peter Bala, Apr 23 2007

A129635 Decimal expansion of 1 + A117871.

Original entry on oeis.org

2, 6, 9, 1, 7, 9, 9, 2, 0, 9, 8, 2, 1, 7, 1, 2, 3, 5, 1, 3, 3, 9, 2, 6, 1, 8, 0, 6, 7, 8, 7, 6, 3, 1, 8, 6, 9, 8, 2, 3, 6, 9, 3, 7, 6, 2, 9, 2, 5, 8, 1, 9, 1, 3, 4, 5, 5, 6, 9
Offset: 1

Views

Author

Jonathan Vos Post, May 31 2007

Keywords

Examples

			2.6917992098217123513392618067876318698236937629258191345569...

Crossrefs

Cf. A092143. See A117871 for a better version.

Extensions

More terms from R. J. Mathar, Sep 02 2007
Edited by N. J. A. Sloane, Sep 16 2007 and May 06 2008

A345726 a(n) = Product_{k=1..n} k^(floor(n/k)^2).

Original entry on oeis.org

1, 2, 6, 192, 960, 4976640, 34836480, 2283043553280, 4993016251023360, 3195530400654950400000, 35150834407204454400000, 417877827219530751882239882035200000, 5432411753853899774469118466457600000, 213700126654516647665669790727613605478400000
Offset: 1

Views

Author

Vaclav Kotesovec, Jun 25 2021

Keywords

Links

Crossrefs

Cf. A073002, A092143, A129364.

Programs

  • Magma
    [(&*[j^(Floor(n/j))^2: j in [1..n]]): n in [1..30]]; // G. C. Greubel, Feb 05 2024
  • Mathematica
    Table[Product[k^(Floor[n/k]^2), {k, 1, n}], {n, 1, 15}]
  • PARI
    a(n) = prod(k=1, n, k^((n\k)^2)); \\ Michel Marcus, Jun 26 2021
  • SageMath
    [product(j^((n//j)^2) for j in range(1,n+1)) for n in range(1,31)] # G. C. Greubel, Feb 05 2024

Formula

log(a(n)) ~ c * n^2, where c = -zeta'(2) = A073002.
Previous Showing 11-15 of 15 results.

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

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