A334360 Anti-Waring numbers: least number k such that k and all larger numbers can be expressed as the sum of one or more distinct n-th powers.

129, 12759, 5134241, 67898772, 11146309948

Offset: 2

Author

Charles R Greathouse IV, Apr 24 2020

Keywords

• nonn
• hard
• more

Comments

Sprague finds a(2) in 1948 and proves that a(n) exists for all n >= 2 in the same year. Graham finds a(3) in 1964 with a paper "to appear" with details; Dressler & Parker give an independent proof in 1974. Lin finds a(4) in 1970. Patterson finds a(5) in 1992. Fuller & Nichols, Jr. find a(6) in 2020.

Examples

			129 = 2^2 + 5^2 + 10^2, but no subset of {1^2, 2^2, ..., 11^2} sums to 128, so a(2) >= 129.
a(3) = 5^3 + 6^3 + 7^3 + 11^3 + 14^3 + 20^3, but a(3) - 1 = 12758 cannot be so represented.
a(4) = 2^4 + 6^4 + 7^4 + 14^4 + 28^4 + 46^4
a(5) = 2^5 + 3^5 + 6^5 + 8^5 + 9^5 + 10^5 + 13^5 + 14^5 + 19^5 + 22^5 + 27^5 + 29^5 + 30^5
		

References

• S. Lin, Computer experiments on sequences which form integral bases, in J. Leech, ed., Computational Problems in Abstract Algebra, Pergamon Press, 1970, pp. 365-370.

Links

• R. Dressler and T. Parker, 12,758, Mathematics of Computation 28:125 (1974), pp. 313-314.
• Chris Fuller and Robert H. Nichols, Jr., Generalized anti-Waring numbers, Journal of Integer Sequences 18 (2015), Article 15.10.5.
• R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964), pp. 275-285.
• C. Patterson, The Derivation of a High Speed Sieve Device, Ph.D. thesis, University of Calgary, 1992. [See 2.2.3.2, Complete Sequences, pp. 18-23.]
• R. Sprague, Über Zerlegung in ungleiche Quadratzahlen, Mathematische Zeitschrift 51 (1948), pp. 289-290.
• R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Mathematische Zeitschrift, 51 (1948), pp. 466-468.

Crossrefs

Cf. A001661.

Programs

PARI

sumOf(n,k,e,xmax=n)=my(t); if(k==1, my(t); if(ispower(n,e,&t) && t<=xmax, return([t]), return(0))); xmax=min(sqrtnint(n,e),xmax); forstep(x=xmax,k,-1, t=sumOf(n-x^e,k-1,e,x-1); if(t, return(concat(t,x)))); 0
bestPowerRep(n,e)=my(k,t); while((t=sumOf(n,k++,e))==0,); t \\ Finds a representation for n as a sum of distinct e-th powers; Charles R Greathouse IV, May 04 2020

Formula

a(n) = A001661(n) + 1. - Ilya Gutkovskiy, Mar 24 2022