# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a001661 Showing 1-1 of 1 %I A001661 M5393 N2342 #102 Jun 10 2023 16:26:17 %S A001661 128,12758,5134240,67898771,11146309947,766834015734,4968618780985762 %N A001661 Largest number not the sum of distinct positive n-th powers. %C A001661 a(8) > 74^8. - _Donovan Johnson_, Nov 23 2010 %C A001661 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 A001661 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 A001661 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 A001661 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 A001661 Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316. %D A001661 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A001661 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A001661 R. E. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313-314. %H A001661 Shalosh B. Ekhad and Doron Zeilberger, Automating John P. D'Angelo's method to study Complete Polynomial Sequences, arXiv:2111.02832 [math.NT], 2021. %H A001661 Mauro Fiorentini, Rappresentazione di interi come somma di potenze (in Italian). %H A001661 C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5. %H A001661 R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964), pp. 275-285. %H A001661 D. Kim, On the largest integer that is not a sum of distinct nth powers of positive integers, arXiv:1610.02439 [math.NT], 2016-2017. %H A001661 D. Kim, On the largest integer that is not a sum of distinct nth powers of positive integers, J. Int. Seq. 20 (2017), #17.7.5. %H A001661 P. LeVan and D. Prier, Improved Bounds on the Anti-Waring Number, J. Int. Seq. 20 (2017), #17.8.7. %H A001661 D. C. Mayer, Sharp bounds for the partition function of integer sequences, BIT 27 (1987), 98-110. %H A001661 D. C. Mayer, Partition functions via bit list operations, 2009. %H A001661 N. J. A. Sloane and R. E. Dressler, Correspondence, June 1974 %H A001661 R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468. %H A001661 Eric Weisstein's World of Mathematics, Waring's Problem %H A001661 M. J. Wiener, The Largest Integer Not the Sum of Distinct 8th Powers, J. Integer Sequences, 26 (2023), Article 23.5.4. %H A001661 J. W. Wrench, Jr., Letter to N. J. A. Sloane, 10 Apr, 1974 %F A001661 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 A001661 a(n) >= (A030052(n)-1)^n. - _M. F. Hasler_, May 15 2020 %Y A001661 Cf. A030052, A173563, A279529. %Y A001661 Cf. A121571 (primes instead of integers). %K A001661 nonn,nice,more,hard %O A001661 2,1 %A A001661 _N. J. A. Sloane_ and _Robert G. Wilson v_ %E A001661 a(7) from _Donovan Johnson_, Nov 23 2010 %E A001661 a(8) from _Michael J. Wiener_, Jun 10 2023 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE