A001661 Largest number not the sum of distinct positive n-th powers.

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

Offset: 2

Author

N. J. A. Sloane and Robert G. Wilson v

Keywords

• nonn
• nice
• more
• hard

Comments

a(8) > 74^8. - Donovan Johnson, Nov 23 2010

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

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

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

References

• 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.
• Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• R. E. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313-314.
• Shalosh B. Ekhad and Doron Zeilberger, Automating John P. D'Angelo's method to study Complete Polynomial Sequences, arXiv:2111.02832 [math.NT], 2021.
• Mauro Fiorentini, Rappresentazione di interi come somma di potenze (in Italian).
• C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5.
• R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964), pp. 275-285.
• 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.
• 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.
• P. LeVan and D. Prier, Improved Bounds on the Anti-Waring Number, J. Int. Seq. 20 (2017), #17.8.7.
• D. C. Mayer, Sharp bounds for the partition function of integer sequences, BIT 27 (1987), 98-110.
• D. C. Mayer, Partition functions via bit list operations, 2009.
• N. J. A. Sloane and R. E. Dressler, Correspondence, June 1974
• R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.
• Eric Weisstein's World of Mathematics, Waring's Problem
• M. J. Wiener, The Largest Integer Not the Sum of Distinct 8th Powers, J. Integer Sequences, 26 (2023), Article 23.5.4.
• J. W. Wrench, Jr., Letter to N. J. A. Sloane, 10 Apr, 1974

Crossrefs

Cf. A030052, A173563, A279529.

Cf. A121571 (primes instead of integers).

Formula

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

a(n) >= (A030052(n)-1)^n. - M. F. Hasler, May 15 2020

Extensions

a(7) from Donovan Johnson, Nov 23 2010

a(8) from Michael J. Wiener, Jun 10 2023