A174420 Let A(n) = floor((3/2)^n), B(n)=3^n-2^n*A(n); then a(n)=2^n-A(n)-B(n)-2.

-2, -2, -1, 0, 8, 4, 26, 98, 68, 245, 284, 941, 908, 2921, 866, 3038, 9773, 95842, 26864, 82811, 776048, 235984, 715436, 2157533, 14878043, 27882168, 16575521, 116892244, 82326503, 515542801, 1009949246, 882651721, 500902958, 1503356036, 4511038850, 13534572662, 40605902153, 53101505973, 21870478820, 65618808017

Offset

0,1

Comments

Related to Waring's problem.

a(n) is conjectured to be >= 0 for all n >= 3. If this were proved it would settle Waring's problem (see A002804). It is known that a(n) >= 0 for 3 <= n <= 471600000.

If we rewrite the formula as (2^n-1)*ceiling((3/2)^n) - 3^n - 1, we see more clearly a comparison between 3^n and the product of an undervaluation of 2^n and an overvaluation of (3/2)^n. If the undervaluation is proportionately smaller than the ceiling overvaluation, a(n) is nonnegative. 2^n grows faster than (3/2)^n, so for a negative value to occur the target difference between (3/2)^n and ceiling((3/2)^n) gets smaller as n gets larger, and the sum of these target differences (for n > 0) is finite. - Peter Munn, Dec 08 2022

References

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 337.

R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.

Links

Table of n, a(n) for n=0..39.

M. Waldschmidt, Open Diophantine problems

Mathematica

a[n_] := Floor[(3/2)^n]; b[n_] := 3^n - 2^n*a[n]; Table[2^n - a[n] - b[n] - 2, {n, 0, 39}] (* Jean-François Alcover, Apr 05 2011 *)

Prog

(Python)
def A174420(n): return ((m:=3**n)&-(k:=1<<n))+k-m-(m>>n)-2 # Chai Wah Wu, Jun 25 2024

Crossrefs

Cf. A002804.

Sequence in context: A327365 A093729 A113080 * A360604 A266318 A011265

Adjacent sequences: A174417 A174418 A174419 * A174421 A174422 A174423

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2010

Status

approved