A260042 Numbers k such that (4^k-1)/3 is not squarefree.

9, 10, 18, 20, 21, 27, 30, 36, 40, 42, 45, 50, 54, 55, 60, 63, 68, 70, 72, 78, 80, 81, 84, 90, 99, 100, 105, 108, 110, 117, 120, 126, 130, 135, 136, 140, 144, 147, 150, 153, 155, 156, 160, 162, 165, 168, 170, 171, 180, 182, 189, 190, 198, 200, 204, 207, 210

Offset: 1

Author

N. J. A. Sloane, Jul 22 2015

Keywords

• nonn

Comments

Contains all positive multiples of 9 (A008591), because 4^n-1 == 0 (mod 27) for these and (4^n-1)/3 is a multiple of 3^2 then. Contains also all positive multiples of 10 (A008592), because 4^n-1 == 0 (mod 125) for these and (4^n-1)/3 is a multiple of 5^2 then. Contains all positive multiples of 21 (A008603), because 4^n-1 == 0 (mod 147) for these and (4^n-1)/3 is a multiple of 7^2 then. - R. J. Mathar, Aug 02 2015

Complement of A259178. - Omar E. Pol, Aug 03 2015

Examples

			(4^9-1)/3 = 3^2*7*19*73 is not squarefree, so 9 is in the sequence. - _R. J. Mathar_, Aug 02 2015
		

References

• James R. Buddenhagen, Posting to Math Fun Mailing List, Jul 22 2015.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..169

Crossrefs

Cf. A002450, A005117, A013929, A259178.

Programs

Magma

[n: n in [1..120]| not IsSquarefree((4^n-1) div 3)]; // Vincenzo Librandi, Jul 27 2015

Mathematica

Select[Range[120],!SquareFreeQ[(4^#-1)/3]&] (* Ivan N. Ianakiev, Jul 23 2015 *)

PARI

isok(k) = !issquarefree((4^k-1)/3); \\ Michel Marcus, Feb 25 2021

Extensions

a(24)-a(31) from Ivan N. Ianakiev, Jul 23 2015

a(32)-a(45) from Chai Wah Wu, Jul 26 2015

a(46)-a(57) from Lars Blomberg, Aug 06 2017