A344034 -id:A344034 - cp's OEIS frontend

A343704 Numbers that are the sum of five positive cubes in three or more ways.

766, 810, 827, 829, 865, 883, 981, 1018, 1025, 1044, 1070, 1105, 1108, 1142, 1145, 1161, 1168, 1226, 1233, 1252, 1259, 1289, 1350, 1368, 1376, 1424, 1431, 1439, 1441, 1457, 1461, 1487, 1492, 1494, 1522, 1529, 1531, 1538, 1548, 1550, 1555, 1568, 1583, 1585, 1587, 1590, 1592, 1593, 1594, 1609, 1611, 1613, 1639

Offset: 1

David Consiglio, Jr., Apr 26 2021

This sequence differs from A343705 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 this sequence but not A343705.

			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.

David Consiglio, Jr., Table of n, a(n) for n = 1..10000

Cf. A025407, A343702, A343705, A344034, A344243, A344796, A345512.

Select[Range@2000,Length@Select[PowersRepresentations[#,5,3],FreeQ[#,0]&]>2&] (* 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])

A343971 Numbers that are the sum of four positive cubes in four or more ways.

Original entry on oeis.org

1979, 2737, 3663, 4384, 4445, 4474, 4949, 5105, 5131, 5257, 5320, 5473, 5499, 5553, 5616, 5733, 5768, 5833, 5852, 5859, 6064, 6104, 6328, 6372, 6435, 6587, 6643, 6832, 6883, 6912, 6974, 7000, 7030, 7120, 7217, 7371, 7560, 7686, 7777, 7840, 8099, 8108, 8281, 8316, 8344, 8379, 8414, 8505, 8568, 8927, 9016, 9018

Offset: 1

David Consiglio, Jr., May 05 2021

nonn

			3663 = 1^3 + 10^3 + 11^3 + 11^3
     = 2^3 +  4^3 +  6^3 + 15^3
     = 2^3 +  9^3 +  9^3 + 13^3
     = 4^3 +  7^3 +  8^3 + 14^3
so 3663 is a term.

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

Cf. A025369, A025407, A343968, A343972, A343987, A344034, A344352.

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,4):
    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])

A343989 Numbers that are the sum of five positive cubes in five or more ways.

Original entry on oeis.org

1765, 1980, 2043, 2104, 2195, 2250, 2430, 2449, 2486, 2491, 2493, 2547, 2584, 2592, 2738, 2745, 2764, 2817, 2888, 2915, 2953, 2969, 2979, 3095, 3096, 3133, 3142, 3186, 3188, 3214, 3240, 3249, 3275, 3277, 3310, 3312, 3366, 3403, 3422, 3459, 3464, 3466, 3483, 3492, 3520, 3529, 3583, 3608, 3627, 3653, 3664, 3671

Offset: 1

David Consiglio, Jr., May 06 2021

nonn

			2043 = 1^3 + 4^3 + 5^3 +  5^3 + 12^3
     = 2^3 + 2^3 + 3^3 + 10^3 + 10^3
     = 2^3 + 3^3 + 4^3 +  6^3 + 12^3
     = 4^3 + 5^3 + 5^3 +  9^3 + 10^3
     = 4^3 + 6^3 + 6^3 +  6^3 + 11^3
so 2043 is a term.

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

Cf. A343987, A343988, A344034, A344358, A344798, A345174, A345514.

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 >= 5])
for x in range(len(rets)):
    print(rets[x])

A345513 Numbers that are the sum of six cubes in four or more ways.

Original entry on oeis.org

626, 830, 837, 856, 873, 891, 947, 954, 982, 1008, 1026, 1045, 1052, 1053, 1071, 1094, 1097, 1106, 1109, 1134, 1143, 1150, 1153, 1169, 1172, 1195, 1208, 1227, 1234, 1241, 1253, 1260, 1267, 1278, 1279, 1283, 1286, 1290, 1297, 1316, 1323, 1324, 1358, 1361, 1368

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			830 is a term because 830 = 1^3 + 1^3 + 2^3 + 3^3 + 3^3 + 8^3 = 1^3 + 3^3 + 3^3 + 5^3 + 5^3 + 6^3 = 1^3 + 3^3 + 3^3 + 3^3 + 4^3 + 7^3 = 2^3 + 2^3 + 3^3 + 3^3 + 6^3 + 6^3.

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

Cf. A344034, A344808, A345512, A345514, A345522, A345561, A345766.

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 >= 4])
    for x in range(len(rets)):
        print(rets[x])

A344035 Numbers that are the sum of five positive cubes in exactly four ways.

Original entry on oeis.org

1252, 1376, 1461, 1522, 1548, 1585, 1590, 1646, 1702, 1709, 1737, 1739, 1772, 1798, 1802, 1810, 1864, 1889, 1954, 1987, 2006, 2033, 2081, 2096, 2152, 2160, 2225, 2241, 2251, 2276, 2313, 2322, 2339, 2341, 2367, 2374, 2377, 2416, 2423, 2456, 2458, 2465, 2467, 2512, 2521, 2528, 2530, 2537, 2540, 2549, 2556, 2582

Offset: 1

David Consiglio, Jr., May 07 2021

nonn

Differs from A344034 at term 13 because 1765 = 1^3 + 1^3 + 2^3 + 3^3 + 12^3 = 1^3 + 1^3 + 6^3 + 6^3 + 11^3 = 1^3 + 2^3 + 3^3 + 9^3 + 10^3 = 3^3 + 4^3 + 6^3 + 9^3 + 9^3 = 4^3 + 4^3 + 5^3 + 8^3 + 10^3

			1461 is a member of this sequence because 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

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

Cf. A294738, A343705, A343972, A343988, A344034, A344355, A345766.

s5pcQ[n_]:=Length[Select[PowersRepresentations[n,5,3],FreeQ[#,0]&]]==4; Select[Range[ 3000],s5pcQ] (* Harvey P. Dale, Sep 15 2024 *)

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])

A344354 Numbers that are the sum of five fourth powers in four or more ways.

Original entry on oeis.org

20995, 21235, 31250, 41474, 43235, 43250, 43315, 43490, 43859, 45139, 46290, 47570, 51939, 53234, 53299, 54994, 56274, 57379, 57410, 57779, 59329, 59779, 63970, 67010, 67859, 68035, 68290, 71795, 71954, 73730, 73954, 75714, 75794, 77890, 82099, 84499, 86275, 86450, 87730, 92500, 93394, 93474, 93859

Offset: 1

David Consiglio, Jr., May 15 2021

nonn

			31250 is a term of this sequence because 31250 = 2^4 + 2^4 + 4^4 + 7^4 + 13^4 = 2^4 + 3^4 + 6^4 + 6^4 + 13^4 = 4^4 + 6^4 + 7^4 + 9^4 + 12^4 = 5^4 + 5^4 + 10^4 + 10^4 + 10^4.

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

Cf. A344034, A344243, A344352, A344355, A344358, A344518, A345561.

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 >= 4])
for x in range(len(rets)):
    print(rets[x])

A344797 Numbers that are the sum of five squares in four or more ways.

Original entry on oeis.org

53, 56, 59, 61, 62, 64, 67, 68, 70, 71, 72, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 86, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126

Offset: 1

Sean A. Irvine, May 28 2021

nonn

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

Cf. A025369, A294738, A344034, A344796, A344797, A344798, A344808.

cp's OEIS Frontend

A343704 Numbers that are the sum of five positive cubes in three or more ways.

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A343971 Numbers that are the sum of four positive cubes in four or more ways.

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A343989 Numbers that are the sum of five positive cubes in five or more ways.

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A345513 Numbers that are the sum of six cubes in four or more ways.

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A344035 Numbers that are the sum of five positive cubes in exactly four ways.

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A344354 Numbers that are the sum of five fourth powers in four or more ways.

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A344797 Numbers that are the sum of five squares in four or more ways.

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