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%I #8 Sep 08 2022 08:45:50
%S 3,37,4219,53412541,8558698768467667,
%T 219753973828109905009157978671669,
%U 144875427039736839295788447509321763094522140738437260379045751691
%N a(1) = 3; for n > 1, a(n) = smallest k such that a(n-1)^3 + k is a cube.
%C a(8) has 131 decimal digits.
%C From a(2) onward a subsequence of A003215 (centered hexagonal numbers: 3n(n+1)+1, also first differences of A000578). - _Klaus Brockhaus_, Mar 20 2010
%e n = 2: for k = 37, a(1)^3+k = 3^3+37 = 64 = 4^3 is a cube; 37 is the smallest such k, therefore a(2) = 37.
%e n = 4: for k = 53412541, a(3)^3+k = 4219^3+53412541 = 75151448000 = 4220^3 is a cube; 53412541 is the smallest such k, therefore a(4) = 53412541.
%o (Magma) /* inefficient, uses definition */ a:=3; S:=[a]; for n in [2..4] do k:=0; flag:= true; while flag do k+:=1; if IsPower(a^3+k, 3) then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;
%o /* uses formula from _R. J. Mathar_, see A172028 */ [ n eq 1 select 3 else 1+3*Self(n-1)*(Self(n-1)+1): n in [1..8] ]; // _Klaus Brockhaus_, Mar 16 2010
%Y Cf. A172028.
%K nonn
%O 1,1
%A _Vincenzo Librandi_, Jan 23 2010
%E Edited, a(4), a(5) corrected and a(6), a(7) added by _Klaus Brockhaus_, Mar 16 2010
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