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A065964 a(n) is the smallest k such that (k^3 + 1)/(n^3 + 1) is an integer > 1.

%I A065964 #15 Dec 09 2024 22:03:35
%S A065964 3,5,19,49,17,26,295,107,649,153,323,69,145,719,3151,3841,251,597,
%T A065964 6499,362,8821,10165,3527,1399,2981,836,1063,21169,7289,3254,607,9899,
%U A065964 4045,21304,13067,3431,867,803,57799,9183,1601,27527,6159,26459,10993,20538
%N A065964 a(n) is the smallest k such that (k^3 + 1)/(n^3 + 1) is an integer > 1.
%C A065964 a(n) exists because n^3 + 1 divides (n^3 - n^2 + 1)^3 + 1. The set S of n such a(n) = n^3 - n^2 + 1 is S = (2, 3, 4, 7, 9, 15, 16, 19, 21, 22, ...).
%H A065964 Harry J. Smith, <a href="/A065964/b065964.txt">Table of n, a(n) for n = 1..300</a>
%t A065964 Do[k = 1; While[m = (k^3 + 1)/(n^3 + 1); m < 2 || !IntegerQ[m], k++ ]; Print[k], {n, 1, 50} ]
%o A065964 (PARI) a(n) = { my(r=n^3+1, k=n+1); while ((k^3 + 1)%r, k++); k } \\ _Harry J. Smith_, Nov 04 2009
%Y A065964 Cf. A065876.
%K A065964 nonn
%O A065964 1,1
%A A065964 _Benoit Cloitre_, Dec 08 2001
%E A065964 Corrected and extended by _Robert G. Wilson v_, Dec 11 2001