A343705 Numbers that are the sum of five positive cubes in exactly three ways.

766, 810, 827, 829, 865, 883, 981, 1018, 1025, 1044, 1070, 1105, 1108, 1142, 1145, 1161, 1168, 1226, 1233, 1259, 1289, 1350, 1368, 1424, 1431, 1439, 1441, 1457, 1487, 1492, 1494, 1529, 1531, 1538, 1550, 1555, 1568, 1583, 1587, 1592, 1593, 1594, 1609, 1611, 1613, 1639, 1648, 1665, 1672, 1674, 1688, 1707, 1711

Offset: 1

Author

David Consiglio, Jr., Apr 26 2021

Keywords

• nonn
• easy

Comments

This sequence differs from A343704 at term 20 because 1252 = 1^3 + 1^3 + 5^3 + 5^3 + 10^3 = 1^3 + 2^3 + 3^3 + 6^3 + 10^3 = 3^3 + 3^3 + 7^3 + 7^3 + 8^3 = 3^3 + 4^3 + 6^3 + 6^3 + 9^3. Thus this term is in A343704 but not in this sequence.

Comment from D. S. McNeil, May 13 2021: (Start)

If we weaken positive cubes to nonnegative cubes, Deshouillers, Hennecart, and Landreau (2000) give numerical and heuristic evidence that all numbers past 7373170279850 are representable as the sum of 4 nonnegative cubes.

So if they are right, then eventually we can just take some N and represent each of (N-1^3, N-2^3, N-3^3, N-4^3) as the sum of four cubes and then take 1^3, 2^3, 3^3, or 4^3 as our fifth cube, giving at least four 5-cube representations for N.

So it is very likely that the set of numbers representable by the sum of 5 positive cubes in exactly three ways is finite. (End)

It is conjectured that the number of ways of writing N as a sum of 5 positive cubes grows like C(N)*N^(2/3), where C(N) depends on N but is bounded away from zero by an absolute constant (Vaughan, 1981; Vaughan and Wooley, 2002). So the number will exceed 3 as soon as N is large enough, and so it is very likely that this sequence is finite. However, at present this is an open question. - N. J. A. Sloane, May 15 2021 (based on correspondence with Robert Vaughan and Trevor Wooley).

Examples

			827 is a term of this sequence because 827 = 1^3 + 4^3 + 5^3 + 5^3 + 8^3 = 2^3 + 2^3 + 5^3 + 7^3 + 7^3 = 2^3 + 3^3 + 4^3 + 6^3 + 8^3.
		

References

• R. C. Vaughan, The Hardy-Littlewood Method, Cambridge University Press, 1981.
• R. C. Vaughan, Trevor Wooley (2002), Waring's Problem: A Survey. In Michael A. Bennet, Bruce C. Berndt, Nigel Boston, Harold G. Diamond, Adolf J. Hildebrand, Walter Philipp (eds.). Number Theory for the Millennium. III. Natick, MA: A. K. Peters, pp. 301-340.

Links

• David Consiglio, Jr. and Sean A. Irvine, Table of n, a(n) for n = 1..18984
• Jean-Marc Deshouillers, François Hennecart, and Bernard Landreau, 7373170279850, Math. Comp. 69 (2000), pp. 421-439. Appendix by I. Gusti Putu Purnaba.

Crossrefs

Equivalent sequences for 1 way: A048926; 2 ways: A048927; 1 or more ways: A003328; 3 or more ways: A343704.

Cf. A003327.

Programs

Mathematica

Select[Range@2000,Length@Select[PowersRepresentations[#,5,3],FreeQ[#,0]&]==3&] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

Python

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1,50)]#n
for pos in cwr(power_terms,5):#m
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v == 3])#s
for x in range(len(rets)):
    print(rets[x])