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, 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

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: A345417 A144790 A090996 * A368010 A237453 A265754

Adjacent sequences: A309733 A309734 A309735 * A309737 A309738 A309739

Keyword

nonn,base

Author

Rémy Sigrist, Aug 14 2019

Status

approved