The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A216063 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A216063

a(n) is the conjectured highest power of n which has no two identical digits in succession.
(history; published version)

#27 by Charles R Greathouse IV at Tue Sep 18 20:08:52 EDT 2012

STATUS

editing

approved

#26 by Charles R Greathouse IV at Tue Sep 18 20:06:43 EDT 2012

COMMENTS

Question: can the (nonzero) value of any terms be proved?

PROG

a(n)=my(best=0); if(n==14, 76, for(k=1, max(9, 30094\sqrt(log(n)), ), if(isA043096(n^k), best=k)); best ) \\ (conjectural) _Charles R Greathouse IV_, Sep 17 2012

\\ (conjectural) Charles R Greathouse IV, Sep 17 2012

STATUS

approved

editing

Discussion

Tue Sep 18

20:08

Charles R Greathouse IV: Of course it's easy to show a(10) = 1...

This formula is more asymptotically conservative; the other would probably fail for any choice of constant (given enough time). In other words: I expect a(n) * sqrt(log n) to be bounded while it is not obvious that a(n) * log n is bounded.

#25 by Charles R Greathouse IV at Tue Sep 18 19:32:08 EDT 2012

STATUS

editing

approved

#24 by Charles R Greathouse IV at Tue Sep 18 19:32:04 EDT 2012

PROG

a(n)=my(best=0); for(k=1, max(9, 147300\log(n)), if(isA043096(n^k), best=k)); best

\\ _(conjectural) _Charles R Greathouse IV_, Sep 17 2012

STATUS

approved

editing

#23 by Charles R Greathouse IV at Tue Sep 18 09:07:38 EDT 2012

STATUS

editing

approved

#22 by Charles R Greathouse IV at Tue Sep 18 09:07:24 EDT 2012

COMMENTS

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

STATUS

approved

editing

#21 by T. D. Noe at Tue Sep 18 01:57:28 EDT 2012

STATUS

editing

approved

#20 by T. D. Noe at Tue Sep 18 01:56:28 EDT 2012

COMMENTS

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

(Start)

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)

(end)

STATUS

proposed

editing

Discussion

Tue Sep 18

01:57

T. D. Noe: Nice questions!

#19 by Charles R Greathouse IV at Mon Sep 17 23:26:29 EDT 2012

STATUS

editing

proposed

#18 by Charles R Greathouse IV at Mon Sep 17 23:25:44 EDT 2012

COMMENTS

Contribution from ~~~~:

(Start)

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

Question: can the (nonzero) value of any terms be proved?

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)

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); for(k=1, max(9, 147\log(n)), if(isA043096(n^k), best=k)); best

\\ Charles R Greathouse IV, Sep 17 2012

CROSSREFS

Cf. A043096, A216064, A216065, A215236.

Cf. A215236.

STATUS

approved

editing

Discussion

Mon Sep 17

23:26

Charles R Greathouse IV: The program, of course, is conjectural.