login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A287286 a(n) = smallest integer s such that every element of the ring of integers mod t for any t can be written as a sum of s n-th powers. 2
1, 4, 4, 15, 5, 9, 4, 32, 13, 12, 11, 16, 6, 14, 15, 64, 6, 27, 4, 25, 24, 23, 23, 32, 10, 26, 40, 29, 29, 31, 5, 128, 33, 10, 35, 37, 9, 9, 39, 41, 41, 49, 12, 44, 15, 47, 10, 64, 13, 62, 51, 53, 53, 81, 60, 56, 14, 59, 5, 61, 11, 12, 63, 256, 65, 67, 12, 68, 69, 71, 6, 73, 16, 74, 75, 16, 14, 84 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

One needs only check a finite number of values (depending on the power).

See Small's paper in references for precise quantitive information.

a(2) <= 4 follows from Lagrange's four squares theorem.

Differs from A040004 only at k=4. - Andrey Zabolotskiy, Jun 03 2017

REFERENCES

G. H. Hardy and J. E. Littlewood, Some Problems of "Partitio Numerorum" (VIII): The Number Gamma(k) in Waring's Problem, Proc London Math Soc. 28 (1928), 518--542. [G. H. Hardy, Collected Papers. Vols. 1-, Oxford Univ. Press, 1966-; see vol. 1, pp. 406-530.]

Wladyslaw Narkiewicz, Rational Number Theory in the 20th Century: From PNT to FLT, Springer Science & Business Media, 2011, pages 154-155.

LINKS

Table of n, a(n) for n=1..78.

H. Sekigawa and K. Koyama, Nonexistence Conditions of a Solution for the congruence x_1^k + ... + x_s^k = N (mod p^n), Math. Comp. 68 (1999), 1283--1297.

C. Small, Waring's problem mod n, Amer. Math. Monthly 84 (1977), no. 1, 12--25.

R. C. Vaughan and T. D. Wooley, Waring’s problem: a survey, Number Theory for the Millennium, III (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 301-340.

EXAMPLE

a(3) <= 4 states that every element of every ring of integers mod m can be written as a sum of 4 (or fewer) cubes. a(3) >= 4, since in Z/9Z, the cubes are {0,1,8} so that 4 is not the sum of any three cubes in Z/9Z. Hence a(3) = 4.

CROSSREFS

Cf. A079611, A174406, A040004.

Sequence in context: A263797 A174406 A270844 * A271546 A325655 A117187

Adjacent sequences:  A287283 A287284 A287285 * A287287 A287288 A287289

KEYWORD

nonn

AUTHOR

David Covert, May 22 2017

EXTENSIONS

Edited by Andrey Zabolotskiy, Jun 10 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 05:14 EST 2019. Contains 329839 sequences. (Running on oeis4.)