A216063 a(n) is the conjectured highest power of n which has no two identical digits in succession.

126, 133, 63, 32, 26, 27, 42, 33, 1, 16, 15, 11, 76, 15, 26, 19, 18, 8, 1, 45, 38, 19, 12, 16, 30, 22, 11, 21, 1, 16, 16, 11, 12, 11, 13, 10, 23, 10, 1, 22, 19, 6, 18, 25, 23, 11, 10, 6, 1, 6, 8, 20, 14, 17, 11, 13, 14, 13, 1, 15, 14, 17, 21, 16, 16, 9, 4, 11

Offset: 2

Author

V. Raman, Sep 01 2012

Keywords

• nonn
• base

Comments

Contribution from Charles R Greathouse IV, Sep 17 2012: (Start)

a(n) = 0 for infinitely many n; such n have positive density in this sequence. Question: are such n of density 1?

A naive heuristic suggests that there are infinitely many n such that a(n) = 6 but only finitely many a(n) such that a(n) > 6. This suggests a weaker conjecture: this sequence is bounded. (End)

Examples

			3^133 = 2865014852390475710679572105323242035759805416923029389510561523 which has no two adjacent identical digits.
		

Links

• T. D. Noe, Table of n, a(n) for n = 2..1000 (terms 2 to 99 from V. Raman).

Crossrefs

Cf. A043096, A216064, A216065, A215236.

Programs

Mathematica

Table[mx = 0; Do[If[! MemberQ[Differences[IntegerDigits[n^k]], 0], mx = k], {k, 1000}]; mx, {n, 2, 100}] (* T. D. Noe, Sep 17 2012 *)

PARI

isA043096(n)=my(v=digits(n));for(i=2,#v,if(v[i]==v[i-1],return(0)));1
a(n)=my(best=0); if(n==14,76,for(k=1, max(9,94\sqrt(log(n))), if(isA043096(n^k), best=k)); best ) \\ (conjectural) Charles R Greathouse IV, Sep 17 2012