A140746 - OEIS

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

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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