A345843 -id:A345843 - cp's OEIS frontend

A003343 Numbers that are the sum of 9 positive 4th powers.

9, 24, 39, 54, 69, 84, 89, 99, 104, 114, 119, 129, 134, 144, 149, 164, 169, 179, 184, 194, 199, 209, 214, 229, 244, 249, 259, 264, 274, 279, 294, 309, 324, 329, 339, 344, 354, 359, 369, 374, 384, 389, 404, 409, 419, 424, 434, 439, 449, 454, 469, 484, 489, 499, 504

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 02 2020: (Start)
4644 is in the sequence as 4644 = 1^4 + 3^4 + 3^4 + 3^4 + 4^4 + 4^4 + 6^4 + 6^4 + 6^4.
7541 is in the sequence as 7541 = 1^4 + 1^4 + 2^4 + 4^4 + 5^4 + 5^4 + 5^4 + 6^4 + 8^4.
10855 is in the sequence as 10855 = 1^4 + 3^4 + 3^4 + 5^4 + 5^4 + 5^4 + 5^4 + 8^4 + 8^4. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A003332, A003342, A003344, A003354, A345586, A345843.

Select[Range[500], AnyTrue[PowersRepresentations[#, 9, 4], First[#]>0&]&] (* Jean-François Alcover, Jul 18 2017 *)

A345844 Numbers that are the sum of nine fourth powers in exactly two ways.

Original entry on oeis.org

264, 279, 294, 309, 324, 339, 344, 359, 374, 389, 404, 424, 439, 454, 469, 504, 549, 564, 579, 584, 614, 629, 644, 664, 679, 694, 709, 759, 789, 804, 819, 839, 854, 869, 884, 888, 903, 918, 933, 934, 948, 949, 968, 983, 998, 1013, 1014, 1029, 1044, 1048, 1059

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345586 at term 17 because 519 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 4^4 + 4^4 = 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 3^4 + 3^4 + 3^4 + 4^4 = 1^4 + 2^4 + 2^4 + 3^4 + 3^4 + 3^4 + 3^4 + 3^4 + 3^4.

			279 is a term because 279 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 4^4 = 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 2^4 + 3^4 + 3^4 + 3^4.

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

Cf. A345586, A345794, A345834, A345843, A345845, A345854, A346337.

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, 9):
    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])

A345833 Numbers that are the sum of eight fourth powers in exactly one ways.

Original entry on oeis.org

8, 23, 38, 53, 68, 83, 88, 98, 103, 113, 118, 128, 133, 148, 163, 168, 178, 183, 193, 198, 213, 228, 243, 248, 258, 328, 338, 353, 368, 403, 408, 418, 433, 468, 483, 488, 498, 568, 578, 593, 608, 632, 643, 647, 648, 658, 662, 663, 673, 677, 692, 707, 708, 712

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A003342 at term 26 because 263 = 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 3^4 + 3^4 + 3^4 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 4^4.

			23 is a term because 23 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4.

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

Cf. A003342, A345783, A345823, A345834, A345843, A346326.

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, 8):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v == 1])
    for x in range(len(rets)):
        print(rets[x])

A345853 Numbers that are the sum of ten fourth powers in exactly one ways.

Original entry on oeis.org

10, 25, 40, 55, 70, 85, 90, 100, 105, 115, 120, 130, 135, 145, 150, 160, 165, 170, 180, 185, 195, 200, 210, 215, 225, 230, 245, 250, 260, 275, 290, 330, 370, 385, 400, 410, 435, 450, 465, 490, 500, 515, 530, 570, 610, 625, 634, 640, 649, 650, 664, 675, 679

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A003344 at term 30 because 265 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 3^4 + 3^4 + 3^4 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 4^4.

			25 is a term because 25 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4.

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

Cf. A003344, A345803, A345843, A345854, A346346.

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

A346336 Numbers that are the sum of nine fifth powers in exactly one way.

Original entry on oeis.org

9, 40, 71, 102, 133, 164, 195, 226, 251, 257, 282, 288, 313, 344, 375, 406, 437, 468, 493, 499, 524, 555, 586, 617, 648, 679, 710, 735, 766, 797, 828, 859, 890, 921, 977, 1008, 1032, 1039, 1063, 1070, 1094, 1101, 1125, 1132, 1156, 1187, 1218, 1219, 1249, 1250

Offset: 1

David Consiglio, Jr., Jul 13 2021

nonn

Differs from A003354 at term 191 because 4101 = 1^5 + 1^5 + 1^5 + 1^5 + 3^5 + 3^5 + 3^5 + 3^5 + 5^5 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 4^5 + 4^5 + 4^5 + 4^5.

			9 is a term because 9 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.

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

Cf. A003354, A345843, A346326, A346337, A346346.

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, 9):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v == 1])
    for x in range(len(rets)):
        print(rets[x])

A345793 Numbers that are the sum of nine cubes in exactly one way.

Original entry on oeis.org

9, 16, 23, 30, 35, 37, 42, 44, 49, 51, 56, 58, 61, 63, 65, 68, 70, 75, 77, 79, 82, 84, 86, 87, 89, 91, 93, 94, 96, 98, 100, 101, 103, 105, 107, 108, 110, 112, 113, 114, 115, 119, 120, 121, 122, 124, 126, 127, 128, 129, 131, 134, 135, 138, 139, 141, 142, 145

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A003332 at term 18 because 72 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^2 + 1^3 + 4^3 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3.

Likely finite.

			16 is a term because 16 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3.

Sean A. Irvine, Table of n, a(n) for n = 1..176

Cf. A003332, A345783, A345794, A345803, A345843.

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, 9):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v == 1])
    for x in range(len(rets)):
        print(rets[x])

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A003343 Numbers that are the sum of 9 positive 4th powers.

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A345844 Numbers that are the sum of nine fourth powers in exactly two ways.

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A345833 Numbers that are the sum of eight fourth powers in exactly one ways.

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A345853 Numbers that are the sum of ten fourth powers in exactly one ways.

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A346336 Numbers that are the sum of nine fifth powers in exactly one way.

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A345793 Numbers that are the sum of nine cubes in exactly one way.

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