A335113 a(n) is the least k > 0 such that Sum_{j=1..n} j*k^j/(k+n) is integer, for n > 1.

1, 3, 1, 4, 1, 7, 1, 6, 1, 11, 1, 4, 1, 5, 1, 12, 1, 19, 1, 12, 1, 4, 1, 40, 1, 15, 1, 58, 1, 31, 1, 18, 1, 7, 1, 58, 1, 13, 1, 22, 1, 43, 1, 24, 1, 10, 1, 10, 1, 27, 1, 22, 1, 15, 1, 8, 1, 31, 1, 46, 1, 9, 1, 78, 1, 15, 1, 36, 1, 71, 1, 112, 1, 10, 1, 14, 1, 55

Offset

2,2

Comments

a(n) represents the smallest integer solution of the equation (x + 2*x^2 + ... + (n - 1)*x^(n - 1) + n*x^n)/(x + n) = m, where m is any positive integer.

We have a(2*k) = 1 for k > 0 because Sum_{j=1..n} j/(1+n) is equal to n/2. For x > 1, Sum_{j=1..n} j*x^j/(x+n) can be simplified to (x + x^(1+n)*(n*x-n-1))/(n+x)*(x-1)^2). - Giovanni Resta, May 24 2020

Links

Antti Karttunen, Table of n, a(n) for n = 2..3612

Formula

a(2*n) = 1, for n > 0. - Giovanni Resta, May 24 2020

Example

For n = 3, a(3) is the smallest integer k > 0 such that f(k) = (3k^3 + 2k^2 + k)/(k + 3) is an integer. Since f(k) is integer for k = 3, 8, 19, 30, 63, we have a(3) = 3.

Mathematica

f[n_, x_] := Sum[j x^j/(x + n), {j, n}]; a[n_] := Block[{k=1}, While[! IntegerQ@ f[n, k], k++]; k]; a /@ Range[2, 79] (* Giovanni Resta, May 24 2020 *)

Prog

(PARI) A335113(n) = for(k=1, oo, if(!(sum(j=1, n, j*(k^j))%(k+n)), return(k))); \\ Antti Karttunen, Dec 09 2021

Crossrefs

Cf. A335112, A335114.

Sequence in context: A094119 A210622 A364065 * A295276 A301856 A301829

Adjacent sequences: A335110 A335111 A335112 * A335114 A335115 A335116

Keyword

nonn

Author

Marco Ripà, May 23 2020

Extensions

More terms from Giovanni Resta, May 24 2020

Status

approved