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 1-2 of 2 results.

A120114 a(n) = lcm(1, ..., 2n+4)/lcm(1, ..., 2n+2).

Original entry on oeis.org

6, 5, 14, 3, 11, 13, 2, 17, 19, 1, 23, 5, 3, 29, 62, 1, 1, 37, 1, 41, 43, 1, 47, 7, 1, 53, 1, 1, 59, 61, 2, 1, 67, 1, 71, 73, 1, 1, 79, 3, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 11, 1, 5, 254, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Comments

The subdiagonal of A120113 is -a(n).
From Robert Israel, Dec 03 2024: (Start)
a(n) is the product of the primes p such that 2*n + 3 or 2*n + 4 is a power of p.
Thus: a(n) = 1 if and only if neither 2*n + 3 nor 2*n + 4 is in A000961.
if n + 1 = 2^k - 1 is a Mersenne number but not a Mersenne prime, then a(n) = 2;
if n + 1 = 2^k - 1 is a Mersenne prime, then a(n) = 2 * (2^k - 1);
otherwise a(n) is odd. (End)
Conjectures from Davide Rotondo, Dec 02 2024: (Start)
Except for 2, if a(n) is even then a(n)/2 is a Mersenne prime.
If a(n)=1 or a(n)=2 then (n*2)+3 is in A061346, or also, or (n+1) is in A083390. (End)

Links

Crossrefs

Cf. A000961, A099996, A120113.

Programs

  • GAP
    List([0..75],n->Lcm(List([1..2*n+4],i->i))/Lcm(List([1..2*n+2],i->i))); # Muniru A Asiru, Mar 04 2019
  • Magma
    A120114:= func< n | Lcm([1..2*n+4])/Lcm([1..2*n+2]) >;
[A120114(n): n in [0..100]]; // G. C. Greubel, May 05 2023
  • Maple
    f:= proc(n) local t,x,S;
   t:= 1;
   for x from 2*n+3 to 2*n+4 do
     S:= numtheory:-factorset(x);
     if nops(S) = 1 then t:= t*S[1] fi;
   od;
   t
end proc:
map(f, [$0..100]); # Robert Israel, Dec 03 2024
  • Mathematica
    Table[(LCM@@Range[2n+4])/LCM@@Range[2n+2],{n,0,100}] (* Harvey P. Dale, Dec 15 2017 *)
  • SageMath
    def A120114(n):
    return lcm(range(1,2*n+5)) // lcm(range(1,2*n+3))
[A120114(n) for n in range(101)] # G. C. Greubel, May 05 2023

Formula

a(n) = A099996(n+2)/A099996(n+1). - Michel Marcus, May 06 2023

Extensions

More terms from Harvey P. Dale, Dec 15 2017

A379676 For n >= 0, a(n) is the least k >= 2 such that (n + 1)*(2*k + n) / 2 is a triangular number (A000217).

Original entry on oeis.org

3, 7, 4, 15, 7, 5, 10, 31, 13, 6, 16, 12, 19, 10, 7, 63, 25, 11, 28, 22, 8, 15, 34, 21, 37, 16, 40, 9, 43, 20, 46, 127, 14, 21, 18, 10, 55, 25, 15, 19, 61, 26, 64, 45, 11, 30, 70, 44, 73, 31, 21, 55, 79, 35, 12, 70, 22, 36, 88, 18, 91, 40, 34, 255, 31, 13, 100, 19, 28, 24, 106, 92, 109, 46, 29, 78, 25, 14, 118, 91, 121, 51, 124, 63, 42, 55, 35, 39, 133, 43, 15
Offset: 0

Views

Author

Ctibor O. Zizka, Dec 29 2024

Keywords

Comments

Also for n >= 0, a(n) is the least k >= 2 such that the Sum_{i = 0..n} (k + i) is a triangular number (A000217). For k = 0, 1 the Sum is a triangular number for all n. The sequences A076114 and A076116 are for square sum and cube sum.

Examples

			n = 4: the least k >= 2 such that (4 + 1)*(2*k + 4)/2 = 5*k + 10 is a triangular number is k = 7, thus a(4) = 7.
n = 5: the least k >= 2 such that (5 + 1)*(2*k + 5)/2 = 6*k + 15 is a triangular number is k = 5, thus a(5) = 5.

Crossrefs

Cf. A000217, A076114, A076116, A083390.

Programs

  • Mathematica
    a[n_] := Module[{k = 2}, While[! IntegerQ[Sqrt[4*(n + 1)*(2*k + n) + 1]], k++]; k]; Array[a, 100, 0] (* Amiram Eldar, Dec 30 2024 *)
  • PARI
    a(n) = my(k=2); while (!ispolygonal((n + 1)*(2*k + n)/2, 3), k++); k; \\ Michel Marcus, Dec 30 2024

Formula

For i >= 0, a(2^i - 1) = 2^(i + 2) - 1, max. values of a(n).
For i >= 0, a(i*(i + 3)/2) = i + 3, min. values of a(n).
For i >= 1, i is not from A083390, a(2*i) = (3*i + 1).
Showing 1-2 of 2 results.

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