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a(n) is the greatest k > 0 such that Sum_{j=1..n} j*(-k)^j/(k+n) is an integer, for n > 1 and a(n) != n + 1.
2

%I #41 Jan 10 2025 02:01:47

%S 8,69,864,13460,245664,5134283,120961664,3172973814,91735537200,

%T 2898687320177,99396054701280,3676223870321288,145888302945326144,

%U 6183540678620338455,278807536726516683264,13325206564150591272362,672921671625708650943696,35804449718312525179171229

%N a(n) is the greatest k > 0 such that Sum_{j=1..n} j*(-k)^j/(k+n) is an integer, for n > 1 and a(n) != n + 1.

%C a(n) represents the greatest integer solution of the equation (- k + 2*k^2 - ... +/- (n - 1)*k^(n - 1) -/+ n*k^n)/(k + n) = m, where m is any integer, while a(n) is not equal to the trivial solution n + 1 (i.e., a(1) != 2 does not exist even if (- 2)/(2 - 1) = - 2).

%C If we introduce the additional constraint m>0, then the corresponding sequence is 8, 2, 864, ...

%F a(n) = A335112(n) + 2*n.

%e For n = 3, a(3) is the largest integer x > 0 such that f(k) = (- 3*k^3 + 2*k^2 - k)/(k - 3) is an integer. Since f(k) is an integer for k = 1, 2, 4, 5, 6, 9, 14, 25, 36, 69, we have a(3) = 69.

%Y Cf. A335112, A335113.

%K nonn

%O 2,1

%A _Marco RipĂ _, May 23 2020