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Numbers that are the sum of five positive cubes in three or more ways.
+10
8
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
COMMENTS
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.
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]&]>2&] (* 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])
Numbers that are the sum of five fourth powers in two or more ways.
+10
8
260, 275, 340, 515, 884, 1555, 2595, 2660, 2675, 2690, 2705, 2755, 2770, 2835, 2930, 2945, 3010, 3185, 3299, 3314, 3379, 3554, 3923, 3970, 3985, 4050, 4115, 4145, 4160, 4210, 4225, 4290, 4355, 4400, 4465, 4594, 4769, 4834, 5075, 5090, 5155, 5265, 5330, 5395, 5440, 5505, 5570, 5699, 6370, 6545, 6580, 6595, 6610
EXAMPLE
340 = 1^4 + 1^4 + 1^4 + 3^4 + 4^4
= 2^4 + 3^4 + 3^4 + 3^4 + 3^4
so 340 is a term of this sequence.
PROG
(Python)
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 >= 2])
for x in range(len(rets)):
print(rets[x])
Numbers that are the sum of four fourth powers in three or more ways.
+10
8
16578, 43234, 49329, 53218, 54978, 57154, 93393, 106354, 107649, 108754, 138258, 151219, 160434, 168963, 173539, 177699, 178738, 181138, 183603, 185298, 195378, 195859, 196418, 197154, 197778, 201683, 202419, 209763, 211249, 216594, 217138, 223074, 234274, 235554, 235569, 236674, 237249, 237699
EXAMPLE
49329 = 2^4 + 2^4 + 12^4 + 13^4
= 4^4 + 8^4 + 9^4 + 14^4
= 6^4 + 9^4 + 12^4 + 12^4
so 49329 is a term of this sequence.
PROG
(Python)
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, 4):
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])
Numbers that are the sum of five fourth powers in four or more ways.
+10
8
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
EXAMPLE
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.
PROG
(Python)
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])
Numbers that are the sum of six fourth powers in three or more ways.
+10
8
2676, 2851, 2916, 4131, 4226, 4241, 4306, 4371, 4481, 4850, 5346, 5411, 5521, 5586, 5651, 6561, 6611, 6626, 6691, 6756, 6771, 6801, 6821, 6836, 6851, 6866, 6931, 7106, 7235, 7475, 7491, 7666, 7841, 7906, 7971, 8146, 8211, 8321, 8386, 8451, 8531, 8706, 9011
EXAMPLE
2851 is a term because 2851 = 1^4 + 1^4 + 1^4 + 4^4 + 6^4 + 6^4 = 2^4 + 2^4 + 3^4 + 3^4 + 4^4 + 7^4 = 2^4 + 3^4 + 3^4 + 3^4 + 6^4 + 6^4.
PROG
(Python)
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, 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])
Numbers that are the sum of five fifth powers in three or more ways.
+10
7
13124675, 28055699, 50043937, 52679923, 53069024, 55097976, 57936559, 60484744, 62260463, 62445305, 70211956, 73133026, 79401728, 80368962, 84766210, 88512249, 93288865, 98824300, 106993391, 113055482, 117173891, 120968132, 123383875, 126416258, 131106051, 131529588, 132022925
PROG
(Python)
from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**5 for x in range(1, 500)]
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])
Numbers that are the sum of five fourth powers in exactly three ways.
+10
7
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, 21250, 21330, 21364, 21410, 21954, 23139, 23795, 24754, 25810, 26578, 28610, 28930, 29330, 29699
COMMENTS
Differs from A344243 at term 31 because 20995 = 1^4 + 1^4 + 1^4 + 4^4 + 12^4 = 2^4 + 3^4 + 3^4 + 3^4 + 12^4 = 2^4 + 6^4 + 9^4 + 9^4 + 9^4 = 4^4 + 6^4 + 7^4 + 7^4 + 11^4
EXAMPLE
6850 is a member of this sequence because 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
PROG
(Python)
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])
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