A279529 -id:A279529 - cp's OEIS frontend

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

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

Offset: 2

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

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).

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

Cf. A030052, A173563, A279529.

Cf. A121571 (primes instead of integers).

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

a(7) from Donovan Johnson, Nov 23 2010

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

A292740 Indices k such that A292547(k) = 0.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Vaclav Kotesovec, Sep 22 2017

nonn

Complement of A290276.

Conjecture: for k > 212594 there are no more terms in this sequence (tested for k < 63000000).

			3 is in the sequence because A292547(3) = 0
8 is not in the sequence because A292547(8) = -1
201254 is in the sequence because A292547(201254) = 0
212594 is in the sequence because A292547(212594) = 0

Vaclav Kotesovec, Table of n, a(n) for n = 1..54766

Cf. A276517, A279329, A279486, A279529, A290276, A292547.

Cf. A001476, A022555, A031980.

With[{nn = 200}, -1 + Position[#, 0][[All, 1]] &@ CoefficientList[ Series[Product[1 + x^((2 k - 1)^3), {k, 1, Floor[nn^(1/3)/2] + 1}], {x, 0, nn}], x]] (* Michael De Vlieger, Sep 22 2017, after Vaclav Kotesovec at A292547 *)

cp's OEIS Frontend

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

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