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.

Showing 1-5 of 5 results.

A119937 Triangle of numbers related to the spectrum of the hydrogen (H) atom.

Original entry on oeis.org

3, 32, 5, 135, 27, 7, 3456, 756, 256, 81, 3500, 800, 300, 125, 44, 172800, 40500, 16000, 7425, 3456, 1300, 694575, 165375, 67375, 33075, 17199, 8575, 3375, 6272000, 1509200, 627200, 318500, 175616, 98000, 51200
Offset: 2

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

The rational number triangle r(m,n):=A120072(m,n)/A120073(m,n), used to compute the spectral series of the hydrogen atom, is mapped to this nonnegative number triangle by multiplying the least common multiples (LCM) for each row m.

Examples

			[3]; [32,5]; [135,27,7]; [3456,756,256,81]; [3500,800,300,125,44]; ...

Links

Crossrefs

The LCM sequence which has been used here is [4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 6350400, 768398400, ...] = A051418(m) = (A003418(m))^2 = (2*A025555(m-1))^2, m >= 2.
The row sums give A119938.

Formula

a(m,n) = r(m,n)*lcm_{k=1..m-1} seq(r(m,k)) with r(m,n) = 1/n^2 - 1/m^2 = A120072(m,n)/A120073(m,n), m >= 2, n = 1..m-1.

A051538 Least common multiple of {b(1),...,b(n)}, where b(k) = k(k+1)(2k+1)/6 = A000330(k).

Original entry on oeis.org

1, 5, 70, 210, 2310, 30030, 60060, 1021020, 19399380, 19399380, 446185740, 2230928700, 6692786100, 194090796900, 12033629407800, 12033629407800, 12033629407800, 445244288088600, 445244288088600, 18255015811632600
Offset: 1

Views

Author

Asher Auel

Keywords

Comments

Also a(n) = lcm(1,...,(2n+2))/12. - Paul Barry, Jun 09 2006. Proof that this is the same sequence, from Martin Fuller, May 06 2007: k, k+1, 2k+1 are coprime so their lcm is the same as their product. Hence a(n) = lcm{k, k+1, 2k+1 | k=1..n}/6. {k, k+1, 2k+1 | k=1..n} = {1..2n+2 excluding even numbers >n+1}. Adding the higher even numbers to the set doubles the LCM. Hence lcm{k, k+1, 2k+1 | k=1..n}/6 = lcm{1..2n+2}/12.

Examples

			a(4) = lcm(1, 5, 14, 30) = 210.

Links

Crossrefs

Second column of A120101.
Cf. A000330.
Cf. A051542 (LCM), A025555.

Programs

  • Haskell
    a051538 n = a051538_list !! (n-1)
a051538_list = scanl1 lcm $ tail a000330_list
-- Reinhard Zumkeller, Mar 12 2014
  • Magma
    [Lcm([1..2*n+2])/12: n in [1..30]]; // G. C. Greubel, May 03 2023
  • Mathematica
    Table[LCM@@Range[2n+2]/12,{n,30}] (* Harvey P. Dale, Apr 25 2011 *)
  • SageMath
    def A051538(n):
    return lcm(range(1,2*n+3))/12
[A051538(n) for n in range(1,31)] # G. C. Greubel, May 03 2023

Extensions

Corrected by James Sellers
Edited by N. J. A. Sloane, May 06 2007

A051543 Quotients of consecutive values of lcm of first n triangular numbers (A000217).

Original entry on oeis.org

3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 2, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 2, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 3, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Asher Auel

Keywords

Examples

			a(5) = A025555(6)/A025555(5) = 210/30 = 7

Links

Crossrefs

Cf. A025555.
Cf. A051542.

Programs

  • Haskell
    a051543 n = a051542_list !! (n-1)
a051543_list = zipWith div (tail a025555_list) a025555_list
-- Reinhard Zumkeller, Mar 12 2014

Formula

a(n) = A025555(n+1)/A025555(n)

Extensions

Corrected and extended by James Sellers

A120108 Number triangle T(n,k) = lcm(1,..,n+1)/lcm(1,..,k+1).

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 12, 6, 2, 1, 60, 30, 10, 5, 1, 60, 30, 10, 5, 1, 1, 420, 210, 70, 35, 7, 7, 1, 840, 420, 140, 70, 14, 14, 2, 1, 2520, 1260, 420, 210, 42, 42, 6, 3, 1, 2520, 1260, 420, 210, 42, 42, 6, 3, 1, 1, 27720, 13860, 4620, 2310, 462, 462, 66, 33, 11, 11, 1
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Examples

			Triangle begins:
    1;
    2,   1;
    6,   3,  1;
   12,   6,  2,  1;
   60,  30, 10,  5, 1;
   60,  30, 10,  5, 1, 1;
  420, 210, 70, 35, 7, 7, 1;

Links

Crossrefs

First column is A003418(n+1). Second column is A025555. Row sums are A120109. Diagonal sums are A120110. Inverse is A120111.

Programs

  • GAP
    Flat(List([0..10],n->List([0..n],k->Lcm(List([1..n+1],i->i))/Lcm(List([1..k+1],i->i))))); # Muniru A Asiru, Feb 26 2019
  • Magma
    A120108:= func< n,k | Lcm([1..n+1])/Lcm([1..k+1]) >;
[A120108(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 04 2023
  • Maple
    T:= (n,k)-> ilcm(seq(q,q=1..n+1))/ilcm(seq(r,r=1..k+1)):
seq(seq(T(n,k),k=0..n),n=0..10); # Muniru A Asiru, Feb 26 2019
  • Mathematica
    f[n_] := f[n] = LCM @@ Range[n];
T[n_, k_] := f[n+1]/f[k+1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 01 2021 *)
  • SageMath
    def f(n): return lcm(range(1,n+2))
def A120108(n,k):
    return f(n)/f(k)
flatten([[A120108(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 04 2023

Formula

Number triangle T(n,k) = [k<=n]*lcm(1,..,n+1)/lcm(1,..,k+1).

A025556 a(n) = sum of the exponents in the prime factorization of LCM{1,3,6,...,C(n+1,2)}.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 28, 28, 29, 29, 29, 29, 29, 29, 30
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A025528, A025555.

Programs

Formula

a(n) = A001222(A025555(n)). - Sean A. Irvine, Sep 06 2019
a(n) = A025528(n+1)-1. - Pontus von Brömssen, Sep 28 2024

Extensions

More terms from Sean A. Irvine, Sep 06 2019
Showing 1-5 of 5 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.