A140746 Numbers n such that n^2 + 3 is powerful, (i.e., is of the form a^2*b^3, with a>=1, b>=1).

1, 37, 79196, 177833, 5290738, 9667939010

Offset: 1

Author

Lekraj Beedassy, Jul 12 2008

Keywords

• nonn
• more

Comments

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

a(7) <= 457189690981. - Giovanni Resta, Aug 23 2018

Examples

			37 is the sequence since 37^2 + 3 = 1372 = 2^2 * 7^3 is powerful.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 37, pp 14, Ellipses, Paris 2008.

Crossrefs

Cf. A001694 (powerful), A001080, A001081, A117950 (n^2+3).

Programs

Mathematica

powerfulQ[n_] := Min@FactorInteger[n][[All, 2]] > 1; Select[Range[100000], powerfulQ[#^2 + 3] &] (* Amiram Eldar, Aug 22 2018 *)

PARI

isok(n) = vecmin(factor(n^2+3)[,2]) > 1; \\ Michel Marcus, Aug 24 2018

Extensions

a(5) corrected and a(6) removed by Amiram Eldar, Aug 22 2018

a(6) from Giovanni Resta, Aug 23 2018