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
KEYWORD
nonn
AUTHOR
Marco Ripà, May 23 2020
EXTENSIONS
More terms from Giovanni Resta, May 24 2020
STATUS
approved