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A343705 Numbers that are the sum of five positive cubes in exactly three ways. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

REFERENCES

R. C. Vaughan, The Hardy-Littlewood Method, Cambridge University Press, 1981.

Vaughan, R. C.; Wooley, Trevor (2002), Waring's Problem: A Survey. In Bennet, Michael A.; Berndt, Bruce C.; Boston, Nigel; Diamond, Harold G.; Hildebrand, Adolf J.; Philipp, Walter (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, Bernard Landreau, 7373170279850, Math. Comp. 69 (2000), pp. 421-439. Appendix by I. Gusti Putu Purnaba.

EXAMPLE

827 is a member 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.

MATHEMATICA

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

PROG

(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])

CROSSREFS

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

Cf. A003327.

Sequence in context: A329757 A127206 A343704 * A247486 A125109 A234163

Adjacent sequences: A343702 A343703 A343704 * A343706 A343707 A343708

KEYWORD

nonn,easy

AUTHOR

David Consiglio, Jr., Apr 26 2021

STATUS

approved

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Last modified January 27 21:49 EST 2023. Contains 359849 sequences. (Running on oeis4.)