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Largest number not the sum of distinct positive n-th powers.
(Formerly M5393 N2342)
17

%I M5393 N2342 #102 Jun 10 2023 16:26:17

%S 128,12758,5134240,67898771,11146309947,766834015734,4968618780985762

%N Largest number not the sum of distinct positive n-th powers.

%C a(8) > 74^8. - _Donovan Johnson_, Nov 23 2010

%C Fuller and Nichols prove that a(6) = 11146309947 and that 2037573096 positive numbers cannot be written as the sum of distinct 6th powers. - _Robert Nichols_, Sep 09 2017

%C a(8) >= 83^8 ~ 2.25e15 since A030052(8) = 84. Similarly, a(9..15) >= (46^9, 62^10, 67^11, 80^12, 101^13, 94^14, 103^15) ~ (9.2e14, 8.4e17, 1.2e20, 6.9e22, 1.1e26, 4.2e27, 1.6e30), cf. formula. Most often a(n) will be closer to and even larger than A030052(n)^n. - In the literature, a(n)+1 is known as the anti-Waring number N(n,1). - _M. F. Hasler_, May 15 2020

%C a(9..16) > (1.55e17, 1.31e19, 1.64e21, 5.55e23, 1.32e26, 1.37e28, 2.09e30, 9.99e35). - _Michael J. Wiener_, Jun 10 2023

%D S. Lin, Computer experiments on sequences which form integral bases, pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.

%D Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H R. E. Dressler and T. Parker, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0327652-1">12,758</a>, Math. Comp. 28 (1974), 313-314.

%H Shalosh B. Ekhad and Doron Zeilberger, <a href="https://arxiv.org/abs/2111.02832">Automating John P. D'Angelo's method to study Complete Polynomial Sequences</a>, arXiv:2111.02832 [math.NT], 2021.

%H Mauro Fiorentini, <a href="http://www.bitman.name/math/article/57/263/">Rappresentazione di interi come somma di potenze</a> (in Italian).

%H C. Fuller and R. H. Nichols Jr., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Fuller/fuller2.html">Generalized Anti-Waring Numbers</a>, J. Int. Seq. 18 (2015), #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 D. Kim, <a href="https://arxiv.org/abs/1610.02439">On the largest integer that is not a sum of distinct nth powers of positive integers</a>, arXiv:1610.02439 [math.NT], 2016-2017.

%H D. Kim, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Kim/kim6.html">On the largest integer that is not a sum of distinct nth powers of positive integers</a>, J. Int. Seq. 20 (2017), #17.7.5.

%H P. LeVan and D. Prier, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Prier/prier3.html">Improved Bounds on the Anti-Waring Number</a>, J. Int. Seq. 20 (2017), #17.8.7.

%H D. C. Mayer, <a href="https://doi.org/10.1007/BF01937358">Sharp bounds for the partition function of integer sequences</a>, BIT 27 (1987), 98-110.

%H D. C. Mayer, <a href="http://www.algebra.at/BitList.htm">Partition functions via bit list operations</a>, 2009.

%H N. J. A. Sloane and R. E. Dressler, <a href="/A001661/a001661_1.pdf">Correspondence, June 1974</a>

%H R. Sprague, <a href="https://doi.org/10.1007/BF01185779">Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen</a>, Math. Z. 51 (1948) 466-468.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem</a>

%H M. J. Wiener, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Wiener/wiener3.html">The Largest Integer Not the Sum of Distinct 8th Powers</a>, J. Integer Sequences, 26 (2023), Article 23.5.4.

%H J. W. Wrench, Jr., <a href="/A001661/a001661.pdf">Letter to N. J. A. Sloane, 10 Apr, 1974</a>

%F a(n) < d*2^(n-1)*(c*2^n + (2/3)*d*(4^n - 1) + 2*d - 2)^n + c*d, where c = n!*2^(n^2) and d = 2^(n^2 + 2*n)*c^(n-1) - 1, according to Kim [2016-2017]. - _Danny Rorabaugh_, Oct 11 2016

%F a(n) >= (A030052(n)-1)^n. - _M. F. Hasler_, May 15 2020

%Y Cf. A030052, A173563, A279529.

%Y Cf. A121571 (primes instead of integers).

%K nonn,nice,more,hard

%O 2,1

%A _N. J. A. Sloane_ and _Robert G. Wilson v_

%E a(7) from _Donovan Johnson_, Nov 23 2010

%E a(8) from _Michael J. Wiener_, Jun 10 2023