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%I #17 Mar 11 2023 08:42:21
%S 129,12759,5134241,67898772,11146309948
%N 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.
%C 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.
%D 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.
%H R. Dressler and T. Parker, <a href="https://www.ams.org/journals/mcom/1974-28-125/S0025-5718-1974-0327652-1/">12,758</a>, Mathematics of Computation 28:125 (1974), pp. 313-314.
%H Chris Fuller and Robert H. Nichols, Jr., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Fuller/fuller2.html">Generalized anti-Waring numbers</a>, Journal of Integer Sequences 18 (2015), Article 15.10.5.
%H R. L. Graham, <a href="http://www.math.ucsd.edu/~ronspubs/64_04_polynomial.pdf">Complete sequences of polynomial values</a>, Duke Math. J. 31 (1964), pp. 275-285.
%H C. Patterson, <a href="http://hdl.handle.net/1880/46539">The Derivation of a High Speed Sieve Device</a>, Ph.D. thesis, University of Calgary, 1992. [See 2.2.3.2, Complete Sequences, pp. 18-23.]
%H R. Sprague, <a href="https://eudml.org/doc/169077">Über Zerlegung in ungleiche Quadratzahlen</a>, Mathematische Zeitschrift 51 (1948), pp. 289-290.
%H R. Sprague, <a href="https://eudml.org/doc/169088">Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen</a>, Mathematische Zeitschrift, 51 (1948), pp. 466-468.
%F a(n) = A001661(n) + 1. - _Ilya Gutkovskiy_, Mar 24 2022
%e 129 = 2^2 + 5^2 + 10^2, but no subset of {1^2, 2^2, ..., 11^2} sums to 128, so a(2) >= 129.
%e a(3) = 5^3 + 6^3 + 7^3 + 11^3 + 14^3 + 20^3, but a(3) - 1 = 12758 cannot be so represented.
%e a(4) = 2^4 + 6^4 + 7^4 + 14^4 + 28^4 + 46^4
%e 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
%o (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
%o 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
%Y Cf. A001661.
%K nonn,hard,more
%O 2,1
%A _Charles R Greathouse IV_, Apr 24 2020
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