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A216439 Smallest k such that k*n^3 + 1 is an n-th power.

%I A216439 #24 Aug 27 2020 13:35:13
%S A216439 1,1,37,791,95051,111748,2277696793,484679258335,229930796172439,
%T A216439 79792266297612,66954547910007962117,337165646545,
%U A216439 45082285083777592171142467,2379140952844779936142872,60722942736706550906445847537201,78033832840595333890814363993704319
%N A216439 Smallest k such that k*n^3 + 1 is an n-th power.
%H A216439 Robert Israel, <a href="/A216439/b216439.txt">Table of n, a(n) for n = 1..216</a>
%e A216439 a(3) = 37 because 37*3^3 + 1 = 1000 = 10^3.
%p A216439 f:= proc(n) local S,x;
%p A216439   S:= subs(1=n^3+1, map(t -> rhs(op(t)), [msolve(x^n-1, n^3)]));
%p A216439   x:= min(S);
%p A216439   (x^n-1)/n^3
%p A216439 end proc:
%p A216439 f(1):= 1:
%p A216439 map(f, [$1..20]); # _Robert Israel_, Aug 26 2020
%t A216439 a={}; Do[k = 2; While[ !IntegerQ[(k^n - 1)/n^3], k++ ]; AppendTo[a, (k^n-1)/n^3], {n, 1, 20}]; a
%Y A216439 Cf. A076943, A193746.
%K A216439 nonn
%O A216439 1,3
%A A216439 _Michel Lagneau_, Sep 10 2012