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

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

Offset

1,1

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)

References

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

Links

Table of n, a(n) for n=1..7.

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A079681 A086656 A034602 * A238799 A113257 A180820

Adjacent sequences: A175177 A175178 A175179 * A175181 A175182 A175183

Keyword

nonn,hard,more

Author

Michel Lagneau, Mar 01 2010

Extensions

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

a(7) from Amiram Eldar, Feb 23 2024

Status

approved