A027763 Smallest k such that 2^^n is not congruent to 2^^(n-1) mod k, where 2^^n denotes the power tower 2^2^...^2 (in which 2 appears n times).

2, 3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 531441, 1594323, 4782969, 14348907, 43046721, 86093443, 344373773, 688747547, 3486784401

Offset: 1

Author

R. K. Guy

Keywords

• nonn
• more

Comments

This sequence shares many terms with A056637, the least prime of class n-. Note that 3^(n-1) is an upper bound for each term and the upper bound is reached for n=13 and n=14. Are all subsequent terms 3^(n-1)? The Mathematica code uses the TowerMod function in the CNT package, which is described in the book by Bressoud and Wagon. - T. D. Noe, Mar 13 2009

For n=15, n=16, and n=17, the terms are also of the form 3^(n-1), but for n=18 and n=19, the terms are prime. - Wayne VanWeerthuizen, Aug 26 2014

A185816(a(n)) = n. - Reinhard Zumkeller, Sep 02 2014

Prime terms seen up to n=20 are in eleven instances of the form j*a(n-1)+1, for j=2, 4, 6, or 12. Note, though, that a(2)=5 and a(8)=719 are exceptions to this pattern. - Wayne VanWeerthuizen, Sep 06 2014

Examples

			2^^2=2^2=4 and 2^^3=2^2^2=16. We find 4 = 16 (mod k) until k=5. So a(3)=5. - _T. D. Noe_, Mar 13 2009
		

References

• David Bressoud and Stan Wagon, A Course in Computational Number Theory, Key College Pub., 2000, p. 96.
• Stan Wagon, posting to Problem of the Week mailing list, Dec 15 1997.

Links

• D. Bressoud, CNT.m Computational Number Theory Mathematica package.
• Stan Wagon, Putnam Problem Notes
• Eric W. Weisstein, MathWorld: Power Tower

Crossrefs

Cf. A056637, A173927, A185816, A246491.

Programs

Mathematica

Needs["CNT`"]; k=1; Table[While[TowerMod[2,n,k]==TowerMod[2,n-1,k], k++ ]; k, {n,10}] (* T. D. Noe, Mar 13 2009 *)

Extensions

Improved the name and changed the offset because I just prepended a term. - T. D. Noe, Mar 13 2009

Corrected and extended by T. D. Noe, Mar 13 2009

Terms a(15)-a(19) from Wayne VanWeerthuizen, Aug 26 2014

Terms a(20)-a(21) from Wayne VanWeerthuizen, Sep 06 2014