A334588 Distance of closest integer power of 10 to the n-th power of 2.

0, 1, 3, 2, 6, 22, 36, 28, 156, 412, 24, 1048, 3096, 1808, 6384, 22768, 34464, 31072, 162144, 424288, 48576, 1097152, 3194304, 1611392, 6777216, 23554432, 32891136, 34217728, 168435456, 436870912, 73741824, 1147483648, 3294967296, 1410065408, 7179869184, 24359738368

Offset

0,3

Comments

a(n) is a measure for how well powers of 2 may be approximated by powers of 10. It is e.g. relevant to unit conventions for the magnitude of data (e.g. 1 megabyte is approximately 1 mebibyte). The closely related sequence of distances of the closest integer power of 2 to the n-th power of 10 gives a measure for the precision of floating point expressions in computing applications.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..3322

Wikipedia, Floating-point arithmetic

Wikipedia, Mebibyte

Example

For n=2, a(2) = |4-1| = 3.
For n=10, a(10) = min(|2^10-10^3|,|2^10-10^4|) = |1024-1000| = 24.

Maple

a:= n-> (m-> (p-> min(m-10^p, 10^(p+1)-m))(ilog10(m)))(2^n):
seq(a(n), n=0..37);  # Alois P. Heinz, May 07 2020

Mathematica

A[n_] := Min[Abs[(2^n) - 10^Length[IntegerDigits[2^n]]],
  Abs[(2^n) - 10^(Length[IntegerDigits[2^n]] - 1)]]

Prog

(PARI) a(n) = my(i=logint(2^n, 10)); min(abs(10^i-2^n), abs(10*10^i-2^n)); \\ Jinyuan Wang, May 06 2020

Crossrefs

Cf. A000079 (2^n), A011557 (10^n).

Sequence in context: A110768 A140230 A356563 * A025240 A137602 A086550

Adjacent sequences: A334585 A334586 A334587 * A334589 A334590 A334591

Keyword

nonn

Author

Timon S. Gutleb, May 06 2020

Extensions

More terms from Jinyuan Wang, May 06 2020

Status

approved