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

1,2

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

References

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

Links

Table of n, a(n) for n=1..6.

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A015061 A015037 A262786 * A350307 A262646 A063681

Adjacent sequences: A140743 A140744 A140745 * A140747 A140748 A140749

Keyword

nonn,more

Author

Lekraj Beedassy, Jul 12 2008

Extensions

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

a(6) from Giovanni Resta, Aug 23 2018

Status

approved