

A335113


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


2



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
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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*xn1))/(n+x)*(x1)^2).  Giovanni Resta, May 24 2020


LINKS

Table of n, a(n) for n=2..79.


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 *)


CROSSREFS

Cf. A335112, A335114.
Sequence in context: A180251 A094119 A210622 * 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



