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%I #16 Aug 27 2018 02:09:13
%S 1,37,79196,177833,5290738,9667939010
%N Numbers n such that n^2 + 3 is powerful, (i.e., is of the form a^2*b^3, with a>=1, b>=1).
%C Florian Luca proved that this sequence is infinite, by showing that 37*x(7*k) + 98*y(7*k) is in the sequence, where x(k) = A001081(k) and y(k) = A001080(k) are solutions of the Pell equation x^2 - 7*y^2 = 1. The sequence of these numbers is 37, 9667939010, 2524807950507510523, 659360302164952911361460078, ... - _Amiram Eldar_, Aug 22 2018
%C a(7) <= 457189690981. - _Giovanni Resta_, Aug 23 2018
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 37, pp 14, Ellipses, Paris 2008.
%e 37 is the sequence since 37^2 + 3 = 1372 = 2^2 * 7^3 is powerful.
%t powerfulQ[n_] := Min@FactorInteger[n][[All, 2]] > 1; Select[Range[100000], powerfulQ[#^2 + 3] &] (* _Amiram Eldar_, Aug 22 2018 *)
%o (PARI) isok(n) = vecmin(factor(n^2+3)[,2]) > 1; \\ _Michel Marcus_, Aug 24 2018
%Y Cf. A001694 (powerful), A001080, A001081, A117950 (n^2+3).
%K nonn,more
%O 1,2
%A _Lekraj Beedassy_, Jul 12 2008
%E a(5) corrected and a(6) removed by _Amiram Eldar_, Aug 22 2018
%E a(6) from _Giovanni Resta_, Aug 23 2018
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