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A216063
a(n) is the conjectured highest power of n which has no two identical digits in succession.
10
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
2,1
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)
LINKS
V. Raman and T. D. Noe, Table of n, a(n) for n = 2..1000 (V. Raman computed the terms 2 to 99)
EXAMPLE
3^133 = 2865014852390475710679572105323242035759805416923029389510561523 which has no two adjacent identical digits.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
V. Raman, Sep 01 2012
STATUS
approved