A343704 -id:A343704 - cp's OEIS frontend

A025407 Numbers that are the sum of 4 positive cubes in 3 or more ways.

1225, 1521, 1582, 1584, 1738, 1764, 1979, 2009, 2249, 2366, 2415, 2422, 2457, 2459, 2485, 2520, 2539, 2737, 2753, 2763, 2790, 2799, 3008, 3094, 3185, 3187, 3213, 3248, 3276, 3392, 3456, 3458, 3465, 3572, 3582, 3600, 3607, 3626, 3656, 3663, 3717, 3736

Offset: 1

David W. Wilson

nonn

Cf. A025368, A025398, A025405, A025406, A343704, A343971, A344241.

{n: A025457(n) >= 3}. - R. J. Mathar, Jun 15 2018

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

Original entry on oeis.org

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

David Consiglio, Jr., Apr 26 2021

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).

			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.

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.

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.

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

Cf. A003327.

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

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])

A344034 Numbers that are the sum of five positive cubes in four or more ways.

Original entry on oeis.org

1252, 1376, 1461, 1522, 1548, 1585, 1590, 1646, 1702, 1709, 1737, 1739, 1765, 1772, 1798, 1802, 1810, 1864, 1889, 1954, 1980, 1987, 2006, 2033, 2043, 2081, 2096, 2104, 2152, 2160, 2195, 2225, 2241, 2250, 2251, 2276, 2313, 2322, 2339, 2341, 2367, 2374, 2377, 2416, 2423, 2430, 2449, 2456, 2458, 2465, 2467, 2486

Offset: 1

David Consiglio, Jr., May 07 2021

nonn

			1461 = 1^3 + 1^3 + 1^3 + 9^3 +  9^3
     = 1^3 + 1^3 + 4^3 + 4^3 + 11^3
     = 3^3 + 3^3 + 4^3 + 7^3 + 10^3
     = 6^3 + 6^3 + 7^3 + 7^3 +  7^3
so 1461 is a term of this sequence.

David Consiglio, Jr., Table of n, a(n) for n = 1..20000

Cf. A343704, A343971, A343989, A344035, A344354, A344797, A345513.

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)]
for pos in cwr(power_terms,5):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v >= 4])
for x in range(len(rets)):
    print(rets[x])

A344243 Numbers that are the sum of five fourth powers in three or more ways.

Original entry on oeis.org

4225, 6610, 6850, 9170, 9235, 9490, 11299, 12929, 14209, 14690, 14755, 14770, 15314, 16579, 16594, 16659, 16834, 17203, 17235, 17315, 17859, 17874, 17939, 18785, 18850, 18979, 19154, 19700, 19715, 20674, 20995, 21235, 21250, 21330, 21364, 21410, 21954, 23139, 23795, 24754, 25810, 26578, 28610, 28930

Offset: 1

David Consiglio, Jr., May 12 2021

nonn

			6850 = 1^4 + 2^4 + 2^4 + 4^4 + 9^4
     = 2^4 + 3^4 + 4^4 + 7^4 + 8^4
     = 3^4 + 3^4 + 6^4 + 6^4 + 8^4
so 6850 is a term of this sequence.

David Consiglio, Jr., Table of n, a(n) for n = 1..20000

Cf. A342687, A343704, A344238, A344241, A344244, A344354, A345560.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**4 for x in range(1,50)]
for pos in cwr(power_terms,5):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v >= 3])
for x in range(len(rets)):
    print(rets[x])

A345512 Numbers that are the sum of six cubes in three or more ways.

Original entry on oeis.org

221, 254, 369, 411, 443, 469, 495, 502, 576, 595, 600, 626, 648, 658, 684, 704, 711, 720, 739, 746, 753, 760, 765, 767, 772, 774, 779, 786, 793, 811, 818, 828, 830, 835, 837, 844, 854, 856, 863, 866, 873, 874, 880, 884, 886, 891, 892, 893, 899, 905, 910, 919

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			254 is a term because 254 = 1^3 + 1^3 + 1^3 + 1^3 + 4^3 + 4^3 = 1^3 + 1^3 + 1^3 + 2^3 + 3^3 + 5^3 = 2^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3.

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A048931, A343704, A344807, A345511, A345513, A345521, A345560.

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, 1000)]
for pos in cwr(power_terms, 6):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 3])
    for x in range(len(rets)):
        print(rets[x])

A343702 Numbers that are the sum of five positive cubes in two or more ways.

Original entry on oeis.org

157, 220, 227, 246, 253, 260, 267, 279, 283, 286, 305, 316, 323, 342, 344, 361, 368, 377, 379, 384, 403, 410, 435, 440, 442, 468, 475, 487, 494, 501, 523, 530, 531, 549, 562, 568, 586, 592, 594, 595, 599, 602, 621, 625, 640, 647, 657, 658, 683, 703, 710, 712, 719, 729, 731, 738, 745, 752, 759, 764, 766, 771, 773, 778, 785

Offset: 1

David Consiglio, Jr., Apr 26 2021

This sequence differs from A048927:
766 = 1^3 + 1^3 + 2^3 + 3^3 + 9^3
    = 1^3 + 4^3 + 4^3 + 5^3 + 8^3
    = 2^3 + 2^3 + 4^3 + 7^3 + 7^3.
So 766 is a term, but not a term of A048927.

			227 = 1^3 + 1^3 + 1^3 + 2^3 + 6^3
    = 2^3 + 3^3 + 4^3 + 4^3 + 4^3
so 227 is a term of this sequence.

David Consiglio, Jr., Table of n, a(n) for n = 1..20000

Cf. A003328, A025406, A048927, A343704, A344238, A344795, A345511.

Select[Range@1000,Length@Select[PowersRepresentations[#,5,3],FreeQ[#,0]&]>1&] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

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 >= 2])#s
for x in range(len(rets)):
    print(rets[x])

A344796 Numbers that are the sum of five squares in three or more ways.

Original entry on oeis.org

29, 32, 35, 37, 40, 43, 44, 46, 51, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 111, 112

Offset: 1

Sean A. Irvine, May 28 2021

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..20000

Cf. A025368, A294737, A343704, A344795, A344797, A344807.

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A025407 Numbers that are the sum of 4 positive cubes in 3 or more ways.

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

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A344034 Numbers that are the sum of five positive cubes in four or more ways.

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A344243 Numbers that are the sum of five fourth powers in three or more ways.

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A345512 Numbers that are the sum of six cubes in three or more ways.

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A343702 Numbers that are the sum of five positive cubes in two or more ways.

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A344796 Numbers that are the sum of five squares in three or more ways.

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