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a(1) = 1; for n > 1, a(n) = smallest k such that a(n-1)^3 + k is a cube.
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%I #13 Sep 08 2022 08:45:50

%S 1,7,169,86191,22286924017,1490120946485455020919,

%T 6661381305464124918785090511089856547876441

%N a(1) = 1; for n > 1, a(n) = smallest k such that a(n-1)^3 + k is a cube.

%C a(8) has 87 decimal digits.

%C A subsequence of A003215 (centered hexagonal numbers: 3n(n+1)+1, also first differences of A000578). - _Klaus Brockhaus_, Mar 20 2010

%C a(11) has 693 digits and is the last term in the b-file. a(12) has 1386 digits and is too large to include in the b-file. - _Harvey P. Dale_, Jul 31 2019

%H Harvey P. Dale, <a href="/A172027/b172027.txt">Table of n, a(n) for n = 1..11</a>

%F a(n) = 1 + 3*a(n-1)*(a(n-1) + 1). - _Zak Seidov_, Jun 25 2010

%e n = 2: for k = 7, a(1)^3+k = 1^3+7 = 9 = 2^3 is a cube; 7 is the smallest such k, therefore a(2) = 7.

%e n = 4: for k = 86191, a(3)^3+k = 169^3+86191 = 4913000 = 170^3 is a cube; 86191 is the smallest such k, therefore a(4) = 86191.

%t NestList[3#^2+3#+1&,1,6] (* _Harvey P. Dale_, Jul 31 2019 *)

%o (Magma) /* inefficient, uses definition */ a:=1; 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 1 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 1,2

%A _Vincenzo Librandi_, Jan 23 2010

%E Edited and a(7) added by _Klaus Brockhaus_, Mar 16 2010