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Numbers that are the sum of six fourth powers in four or more ways.
+10
8
6626, 6691, 6866, 9251, 9491, 10115, 10706, 10786, 11555, 12595, 14225, 14691, 14771, 15315, 15330, 15395, 15570, 16051, 16595, 16610, 16660, 16675, 16850, 17090, 17091, 17236, 17316, 17331, 17346, 17860, 17875, 17940, 17955, 18195, 18786, 18851, 18866, 19155
EXAMPLE
6691 is a term because 6691 = 1^4 + 1^4 + 1^4 + 6^4 + 6^4 + 8^4 = 1^4 + 2^4 + 2^4 + 2^4 + 3^4 + 9^4 = 2^4 + 2^4 + 3^4 + 3^4 + 7^4 + 8^4 = 3^4 + 4^4 + 4^4 + 6^4 + 7^4 + 7^4.
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 >= 4])
for x in range(len(rets)):
print(rets[x])
Numbers that are the sum of six fifth powers in three or more ways.
+10
7
696467, 893572, 1100264, 1109295, 1165727, 1711776, 2007401, 2025309, 2221767, 2801812, 3047519, 3310494, 3360608, 3550866, 3559556, 3576120, 3807122, 3907101, 4055922, 4093540, 4096114, 4104067, 4123363, 4135578, 4155107, 4195571, 4222339, 4326784, 4417112
EXAMPLE
893572 is a term because 893572 = 2^5 + 6^5 + 7^5 + 12^5 + 12^5 + 13^5 = 2^5 + 7^5 + 7^5 + 11^5 + 11^5 + 14^5 = 5^5 + 8^5 + 8^5 + 8^5 + 8^5 + 15^5.
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, 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 seven fifth powers in four or more ways.
+10
7
893604, 1117071, 1182534, 1414559, 1629244, 1933328, 2280543, 2586035, 2867074, 3050644, 3055295, 3055977, 3256432, 3329360, 3369543, 3436058, 3551890, 3576363, 3896969, 3914877, 3930849, 4055954, 4087746, 4088690, 4093572, 4096665, 4098161, 4104068, 4104310
EXAMPLE
1117071 is a term because 1117071 = 1^5 + 2^5 + 6^5 + 7^5 + 7^5 + 14^5 + 14^5 = 2^5 + 2^5 + 4^5 + 6^5 + 10^5 + 12^5 + 15^5 = 3^5 + 4^5 + 7^5 + 7^5 + 7^5 + 7^5 + 16^5 = 3^5 + 5^5 + 6^5 + 6^5 + 7^5 + 8^5 + 16^5.
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, 1000)]
for pos in cwr(power_terms, 7):
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 fifth powers in five or more ways.
+10
7
54827300, 74115800, 74883600, 75609125, 113088250, 120274275, 166078869, 169692136, 174781858, 178736448, 182341225, 185558208, 194939538, 203054589, 218814275, 235067008, 250989825, 251772882, 252721458, 255453233, 258124975, 274616694, 282859667
EXAMPLE
74115800 is a term because 74115800 = 1^5 + 4^5 + 21^5 + 21^5 + 29^5 + 34^5 = 1^5 + 8^5 + 14^5 + 23^5 + 32^5 + 32^5 = 4^5 + 11^5 + 13^5 + 22^5 + 24^5 + 36^5 = 5^5 + 6^5 + 19^5 + 20^5 + 24^5 + 36^5 = 6^5 + 25^5 + 25^5 + 25^5 + 29^5 + 30^5.
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, 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 >= 5])
for x in range(len(rets)):
print(rets[x])
Numbers that are the sum of six fifth powers in exactly four ways.
+10
7
12047994, 20646208, 21017489, 21300963, 21741819, 24993485, 27669050, 28576064, 30193856, 30785920, 35480456, 35735194, 36082750, 37303264, 39035975, 46814942, 47963291, 50047062, 50724345, 52987561, 53076800, 53606848, 55101101, 56766906, 57969327, 58125980
COMMENTS
Differs from A345718 at term 23 because 54827300 = 4^5 + 7^5 + 21^5 + 22^5 + 23^5 + 33^5 = 5^5 + 10^5 + 15^5 + 20^5 + 28^5 + 32^5 = 1^5 + 14^5 + 16^5 + 19^5 + 28^5 + 32^5 = 4^5 + 11^5 + 13^5 + 22^5 + 29^5 + 31^5 = 5^5 + 6^5 + 19^5 + 20^5 + 29^5 + 31^5.
EXAMPLE
12047994 is a term because 12047994 = 7^5 + 9^5 + 12^5 + 14^5 + 17^5 + 25^5 = 5^5 + 10^5 + 13^5 + 15^5 + 16^5 + 25^5 = 1^5 + 1^5 + 3^5 + 4^5 + 21^5 + 24^5 = 4^5 + 6^5 + 15^5 + 15^5 + 21^5 + 23^5.
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, 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])
Numbers that are the sum of five fifth powers in four or more ways.
+10
6
287618651, 1386406515, 1763135232, 2494769760, 2619898293, 3096064443, 3291315732, 3749564512, 4045994624, 5142310350, 5183605813, 5658934676, 5880926107, 7205217018, 7401155424, 7691215599, 8429499101, 8926086432, 9006349824, 9051501568, 9203796832
EXAMPLE
287618651 is a term because 287618651 = 9^5 + 22^5 + 28^5 + 28^5 + 49^5 = 10^5 + 14^5 + 27^5 + 38^5 + 47^5 = 12^5 + 13^5 + 24^5 + 42^5 + 45^5 = 12^5 + 21^5 + 23^5 + 31^5 + 49^5.
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 >= 4])
for x in range(len(rets)):
print(rets[x])
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