A345822 -id:A345822 - cp's OEIS frontend

A345821 Numbers that are the sum of six fourth powers in exactly nine ways.

88595, 132546, 134931, 144835, 146450, 162355, 170275, 171555, 171795, 172036, 172835, 177380, 177716, 180770, 183540, 184835, 185555, 187700, 187715, 190100, 190211, 193635, 195380, 195780, 196435, 197780, 199075, 199475, 199730, 199955, 202196, 202980

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345566 at term 2 because 122915 = 1^4 + 3^4 + 6^4 + 9^4 + 10^4 + 18^4 = 1^4 + 4^4 + 7^4 + 8^4 + 15^4 + 16^4 = 1^4 + 7^4 + 9^4 + 10^4 + 14^4 + 16^4 = 2^4 + 3^4 + 4^4 + 5^4 + 14^4 + 17^4 = 2^4 + 4^4 + 5^4 + 7^4 + 11^4 + 18^4 = 2^4 + 9^4 + 9^4 + 12^4 + 14^4 + 15^4 = 3^4 + 5^4 + 6^4 + 6^4 + 11^4 + 18^4 = 3^4 + 8^4 + 10^4 + 11^4 + 13^4 + 16^4 = 5^4 + 6^4 + 7^4 + 11^4 + 14^4 + 16^4 = 8^4 + 8^4 + 9^4 + 10^4 + 11^4 + 17^4.

			122915 is a term because 122915 = 1^4 + 3^4 + 6^4 + 9^4 + 10^4 + 18^4 = 1^4 + 4^4 + 7^4 + 8^4 + 15^4 + 16^4 = 1^4 + 7^4 + 9^4 + 10^4 + 14^4 + 16^4 = 2^4 + 3^4 + 4^4 + 5^4 + 14^4 + 17^4 = 2^4 + 4^4 + 5^4 + 7^4 + 11^4 + 18^4 = 2^4 + 9^4 + 9^4 + 12^4 + 14^4 + 15^4 = 3^4 + 5^4 + 6^4 + 6^4 + 11^4 + 18^4 = 3^4 + 8^4 + 10^4 + 11^4 + 13^4 + 16^4 = 5^4 + 6^4 + 7^4 + 11^4 + 14^4 + 16^4 = 8^4 + 8^4 + 9^4 + 10^4 + 11^4 + 17^4.

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

Cf. A341892, A345566, A345771, A345820, A345822, A345831, A346364.

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 == 9])
    for x in range(len(rets)):
        print(rets[x])

A345567 Numbers that are the sum of six fourth powers in ten or more ways.

Original entry on oeis.org

122915, 151556, 161475, 162755, 173075, 183620, 185315, 197795, 199106, 199940, 201875, 201955, 202275, 204275, 204340, 204595, 206115, 207395, 209795, 211075, 212420, 213731, 217620, 217826, 217891, 218515, 221250, 223715, 223955, 224180, 224451, 225875

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			151556 is a term because 151556 = 1^4 + 2^4 + 2^4 + 9^4 + 11^4 + 19^4 = 1^4 + 2^4 + 3^4 + 7^4 + 16^4 + 17^4 = 1^4 + 8^4 + 11^4 + 12^4 + 13^4 + 17^4 = 2^4 + 3^4 + 7^4 + 8^4 + 11^4 + 19^4 = 3^4 + 3^4 + 3^4 + 4^4 + 12^4 + 19^4 = 3^4 + 4^4 + 11^4 + 11^4 + 14^4 + 17^4 = 3^4 + 4^4 + 13^4 + 13^4 + 13^4 + 16^4 = 4^4 + 6^4 + 9^4 + 9^4 + 9^4 + 19^4 = 4^4 + 7^4 + 11^4 + 11^4 + 11^4 + 18^4 = 4^4 + 8^4 + 9^4 + 13^4 + 13^4 + 17^4.

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

Cf. A341897, A344196, A345519, A345566, A345576, A345822.

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 >= 10])
    for x in range(len(rets)):
        print(rets[x])

A345832 Numbers that are the sum of seven fourth powers in exactly ten ways.

Original entry on oeis.org

31251, 44547, 45827, 45892, 47667, 47971, 49572, 51092, 53316, 53476, 54531, 54596, 54756, 57411, 58276, 58660, 59781, 59811, 59827, 59861, 59876, 59892, 61076, 64581, 65876, 65891, 66356, 66596, 66676, 67716, 67876, 68131, 68322, 68772, 69171, 69667, 70116

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345576 at term 5 because 45907 = 1^4 + 1^4 + 3^4 + 4^4 + 8^4 + 12^4 + 12^4 = 1^4 + 6^4 + 6^4 + 8^4 + 8^4 + 9^4 + 13^4 = 2^4 + 2^4 + 2^4 + 4^4 + 7^4 + 11^4 + 13^4 = 2^4 + 2^4 + 3^4 + 6^4 + 6^4 + 11^4 + 13^4 = 2^4 + 2^4 + 4^4 + 7^4 + 7^4 + 7^4 + 14^4 = 2^4 + 3^4 + 6^4 + 6^4 + 7^4 + 7^4 + 14^4 = 2^4 + 4^4 + 6^4 + 7^4 + 9^4 + 11^4 + 12^4 = 2^4 + 5^4 + 5^4 + 10^4 + 10^4 + 10^4 + 11^4 = 3^4 + 3^4 + 4^4 + 4^4 + 4^4 + 9^4 + 14^4 = 3^4 + 6^4 + 6^4 + 6^4 + 9^4 + 11^4 + 12^4 = 4^4 + 7^4 + 7^4 + 8^4 + 8^4 + 8^4 + 13^4.

			44547 is a term because 44547 = 1^4 + 2^4 + 2^4 + 2^4 + 6^4 + 11^4 + 13^4 = 1^4 + 2^4 + 2^4 + 6^4 + 7^4 + 7^4 + 14^4 = 1^4 + 2^4 + 6^4 + 6^4 + 9^4 + 11^4 + 12^4 = 1^4 + 6^4 + 7^4 + 8^4 + 8^4 + 8^4 + 13^4 = 2^4 + 2^4 + 8^4 + 9^4 + 9^4 + 9^4 + 12^4 = 2^4 + 4^4 + 6^4 + 6^4 + 9^4 + 9^4 + 13^4 = 2^4 + 4^4 + 7^4 + 7^4 + 8^4 + 11^4 + 12^4 = 3^4 + 3^4 + 4^4 + 4^4 + 7^4 + 12^4 + 12^4 = 3^4 + 6^4 + 6^4 + 7^4 + 8^4 + 11^4 + 12^4 = 4^4 + 4^4 + 8^4 + 8^4 + 9^4 + 11^4 + 11^4.

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

Cf. A345576, A345782, A345822, A345831, A345842, A346259.

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, 7):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v == 10])
    for x in range(len(rets)):
        print(rets[x])

A341898 Numbers that are the sum of five fourth powers in exactly ten ways.

Original entry on oeis.org

954979, 1205539, 1574850, 1713859, 1863859, 1877394, 1882579, 2071939, 2109730, 2225859, 2288179, 2419954, 2492434, 2495939, 2605314, 2711394, 2784499, 2835939, 2847394, 2880994, 2924674, 3007474, 3061939, 3071379, 3074179, 3117235, 3127219, 3174834, 3190899

Offset: 1

David Consiglio, Jr., Jun 04 2021

nonn

Differs at term 5 because
1801459 =  1^4 +  4^4 +  5^4 + 28^4 + 33^4
        =  1^4 +  4^4 + 12^4 + 23^4 + 35^4
        =  1^4 +  7^4 + 16^4 + 30^4 + 31^4
        =  1^4 + 16^4 + 18^4 + 19^4 + 35^4
        =  3^4 +  6^4 + 18^4 + 21^4 + 35^4
        =  5^4 +  7^4 + 19^4 + 24^4 + 34^4
        =  5^4 +  9^4 + 14^4 + 29^4 + 32^4
        =  7^4 +  9^4 + 16^4 + 25^4 + 34^4
        =  7^4 + 14^4 + 16^4 + 21^4 + 35^4
        =  8^4 +  9^4 + 20^4 + 29^4 + 31^4
        = 10^4 + 19^4 + 19^4 + 21^4 + 34^4.

			954979 =  1^4 +  2^4 + 11^4 + 19^4 + 30^4
       =  1^4 +  7^4 + 18^4 + 25^4 + 26^4
       =  3^4 +  8^4 + 17^4 + 20^4 + 29^4
       =  4^4 +  8^4 + 13^4 + 25^4 + 27^4
       =  4^4 +  9^4 + 10^4 + 11^4 + 31^4
       =  6^4 +  6^4 + 15^4 + 21^4 + 29^4
       =  7^4 + 10^4 + 18^4 + 19^4 + 29^4
       = 11^4 + 11^4 + 20^4 + 22^4 + 27^4
       = 16^4 + 17^4 + 17^4 + 24^4 + 25^4
       = 18^4 + 19^4 + 20^4 + 23^4 + 23^4
so 954979 is a term.

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

Cf. A341892, A341897, A344929, A345188, A345822.

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, 5):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k, v in keep.items() if v == 10])
for x in range(len(rets)):
    print(rets[x])

A345772 Numbers that are the sum of six cubes in exactly ten ways.

Original entry on oeis.org

3215, 3267, 3313, 3339, 3374, 3465, 3493, 3528, 3547, 3584, 3645, 3654, 3698, 3736, 3745, 3752, 3754, 3780, 3789, 3843, 3869, 3878, 3880, 3888, 3906, 3915, 3923, 3950, 3995, 4004, 4014, 4041, 4067, 4122, 4148, 4211, 4212, 4214, 4265, 4266, 4268, 4338, 4349

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345519 at term 1 because 3104 = 1^3 + 2^3 + 7^3 + 8^3 + 8^3 + 12^3 = 1^3 + 5^3 + 5^3 + 5^3 + 10^3 + 12^3 = 2^3 + 2^3 + 4^3 + 4^3 + 6^3 + 14^3 = 2^3 + 3^3 + 4^3 + 7^3 + 11^3 + 11^3 = 2^3 + 3^3 + 5^3 + 6^3 + 10^3 + 12^3 = 2^3 + 7^3 + 8^3 + 8^3 + 9^3 + 10^3 = 3^3 + 3^3 + 5^3 + 6^3 + 8^3 + 13^3 = 4^3 + 5^3 + 7^3 + 8^3 + 9^3 + 11^3 = 5^3 + 5^3 + 5^3 + 9^3 + 10^3 + 10^3 = 5^3 + 6^3 + 6^3 + 6^3 + 10^3 + 11^3 = 6^3 + 6^3 + 6^3 + 6^3 + 8^3 + 12^3.

			3215 is a term because 3215 = 1^3 + 1^3 + 1^3 + 4^3 + 6^3 + 13^3 = 1^3 + 1^3 + 2^3 + 2^3 + 9^3 + 12^3 = 1^3 + 1^3 + 3^3 + 8^3 + 8^3 + 11^3 = 1^3 + 2^3 + 3^3 + 5^3 + 8^3 + 12^3 = 1^3 + 3^3 + 3^3 + 3^3 + 10^3 + 11^3 = 1^3 + 3^3 + 8^3 + 8^3 + 8^3 + 9^3 = 2^3 + 3^3 + 6^3 + 6^3 + 8^3 + 11^3 = 3^3 + 3^3 + 3^3 + 8^3 + 9^3 + 10^3 = 3^3 + 5^3 + 5^3 + 5^3 + 6^3 + 12^3 = 6^3 + 6^3 + 6^3 + 6^3 + 7^3 + 10^3.

Sean A. Irvine, Table of n, a(n) for n = 1..1454

Cf. A345188, A345519, A345771, A345782, A345822.

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 == 10])
    for x in range(len(rets)):
        print(rets[x])

A346365 Numbers that are the sum of six fifth powers in exactly ten ways.

Original entry on oeis.org

55302546200, 89999127392, 96110537743, 104484239200, 120492759200, 121258798144, 127794946400, 133364991375, 135030535200, 136156575744, 151305014432, 155434423925, 174388570400, 177099008000, 179272687000, 182844944832, 184948721056, 187873845500

Offset: 1

David Consiglio, Jr., Jul 18 2021

nonn

This sequence differs from A344196:
180336745600 = 48^5 + 54^5 +  66^5 +  66^5 + 112^5 + 174^5
             =  9^5 + 21^5 +  93^5 + 112^5 + 117^5 + 168^5
             = 11^5 + 44^5 +  73^5 +  92^5 + 133^5 + 167^5
             = 15^5 + 81^5 +  94^5 +  95^5 + 129^5 + 166^5
             =  1^5 + 49^5 +  62^5 + 107^5 + 138^5 + 163^5
             = 35^5 + 69^5 +  75^5 +  98^5 + 141^5 + 162^5
             = 18^5 + 81^5 + 105^5 + 112^5 + 135^5 + 159^5
             = 14^5 + 50^5 +  62^5 +  86^5 + 150^5 + 158^5
             =  2^5 + 52^5 +  54^5 + 108^5 + 146^5 + 158^5
             = 14^5 + 22^5 +  66^5 + 118^5 + 142^5 + 158^5
             =  4^5 + 50^5 +  58^5 + 102^5 + 150^5 + 156^5,
so 180336745600 is in A344196, but is not in this sequence.

			55302546200 = 34^5 + 38^5 + 50^5 + 57^5 + 95^5 + 136^5
            = 23^5 + 49^5 + 61^5 + 69^5 + 107^5 + 131^5
            = 24^5 + 37^5 + 63^5 + 81^5 + 104^5 + 131^5
            = 21^5 + 35^5 + 60^5 + 94^5 + 100^5 + 130^5
            = 57^5 + 60^5 + 71^5 + 75^5 + 109^5 + 128^5
            = 19^5 + 37^5 + 56^5 + 96^5 + 104^5 + 128^5
            = 35^5 + 41^5 + 53^5 + 69^5 + 115^5 + 127^5
            = 16^5 + 49^5 + 53^5 + 83^5 + 112^5 + 127^5
            = 35^5 + 37^5 + 40^5 + 88^5 + 119^5 + 121^5
            = 11^5 + 24^5 + 71^5 + 104^5 + 109^5 + 121^5
so 55302546200 is a term.

Sean A. Irvine, Table of n, a(n) for n = 1..57

Cf. A344196, A345822, A346259, A346364.

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 == 10])
    for x in range(len(rets)):
        print(rets[x])

cp's OEIS Frontend

A345821 Numbers that are the sum of six fourth powers in exactly nine ways.

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A345567 Numbers that are the sum of six fourth powers in ten or more ways.

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A345832 Numbers that are the sum of seven fourth powers in exactly ten ways.

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A341898 Numbers that are the sum of five fourth powers in exactly ten ways.

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A345772 Numbers that are the sum of six cubes in exactly ten ways.

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A346365 Numbers that are the sum of six fifth powers in exactly ten ways.

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