A175180 Numbers k such that k^2 + 2 is powerful in the sense of A001694.

5, 265, 13775, 716035, 9980583, 37220045, 1934726305

Offset: 1

Author

Michel Lagneau, Mar 01 2010

Keywords

• nonn
• hard
• more

Comments

This sequence is infinite (F. Luca in De Koninck).

The values of k^2 are a subset of A076445, so 23 terms of the sequence are known from there. - R. J. Mathar, Mar 05 2010

Together with 1, supersequence of A238799. - Arkadiusz Wesolowski, Mar 06 2014

From Amiram Eldar, Feb 23 2024: (Start)

a(8) <= 100568547815.

A041042(2*k) is a term for all k >= 0 (since 3^3 * A041043(n)^2 - A041042(n)^2 = -1 if n is odd and 2 if n is even). (End)

Examples

			5 is in the sequence because 5^2 + 2 = 3^3 is powerful.
265 is in the sequence because 265^2 + 2 = 51^2*3^3 is powerful.
13775 is in the sequence because 13775^2 + 2 = 2651^2 * 3^3 is powerful.
		

References

• Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 265, p. 71, Ellipses, Paris, 2008.

Links

• Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 66.

Crossrefs

Cf. A001694, A041042, A041043, A076445, A238799.

Programs

Mathematica

q[n_] := AllTrue[FactorInteger[n^2+2][[;;, 2]], # > 1 &]; Select[Range[10^6], q] (* Amiram Eldar, Feb 23 2024 *)

PARI

is(n)=ispowerful(n^2+2) \\ Charles R Greathouse IV, Feb 04 2013

Extensions

Examples rephrased by R. J. Mathar, Feb 24 2010, Mar 05 2010

a(7) from Amiram Eldar, Feb 23 2024