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A309736 a(1) = 1, and for any n > 1, a(n) is the least k > 0 such that the binary representation of n^k starts with "10". 1
1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 5, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 10, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The sequence is well defined; for any n > 0:

- if n is a power of 2, then a(n) = 1,

- if n is not a power of 2, then log_2(n) is irrational,

  hence the function k -> frac(k * log_2(n)) is dense in the interval [0, 1]

  according to Weyl's criterion,

  so for some k > 0, k*log_2(n) = m + 1 + e where m is a positive integer

                                              and 0 <= e < log_2(3) - 1 < 1,

- hence 2 * 2^m <= n^k < 3 * 2^m and a(n) <= k, QED.

LINKS

Table of n, a(n) for n=1..87.

Rémy Sigrist, Scatterplot of (x, y) such that the binary representation of x^y starts with "10" and x = 2..1024 and y = 1..1024

Eric Weisstein's World of Mathematics, Weyl's Criterion

FORMULA

a(n) = 1 iff n belongs to A004754.

a(2*n) = a(n).

A090996(n^a(n)) = 1.

EXAMPLE

For n = 7:

- the first powers of 7, in decimal as well as in binary, are:

    k  7^k  bin(7^k)

    -  ---  ---------

    1    7        111

    2   49     110001

    3  343  101010111

- hence a(7) = 3.

PROG

(PARI) a(n) = { my (nk=n); for (k=1, oo, if (binary(2*nk)[2]==0, return (k), nk *= n)) }

CROSSREFS

Cf. A004754, A090996, A098174.

Sequence in context: A159864 A144790 A090996 * A237453 A265754 A089309

Adjacent sequences:  A309733 A309734 A309735 * A309737 A309738 A309739

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Aug 14 2019

STATUS

approved

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Last modified July 11 21:30 EDT 2020. Contains 335652 sequences. (Running on oeis4.)