A345541 -id:A345541 - cp's OEIS frontend

A345542 Numbers that are the sum of nine positive cubes in three or more ways.

224, 231, 238, 245, 250, 257, 259, 264, 271, 276, 278, 280, 283, 285, 287, 290, 292, 294, 297, 299, 301, 302, 309, 311, 313, 315, 316, 318, 320, 322, 327, 334, 335, 337, 339, 341, 346, 348, 350, 353, 355, 357, 362, 365, 372, 374, 376, 379, 381, 383, 386, 387

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			231 is a term because 231 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 5^3 = 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 3^3 + 3^3 + 3^3 + 3^3 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3.

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

Cf. A345500, A345533, A345541, A345543, A345551, A345587, A345795.

Select[Range[400],Length[Select[PowersRepresentations[#,9,3],FreeQ[ #,0]&]]> 2&] (* Harvey P. Dale, Jan 04 2022 *)

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 >= 3])
    for x in range(len(rets)):
        print(rets[x])

Definition corrected by Harvey P. Dale, Jan 04 2022

A345586 Numbers that are the sum of nine fourth powers in two or more ways.

Original entry on oeis.org

264, 279, 294, 309, 324, 339, 344, 359, 374, 389, 404, 424, 439, 454, 469, 504, 519, 534, 549, 564, 579, 584, 599, 614, 629, 644, 664, 679, 694, 709, 759, 774, 789, 804, 819, 839, 854, 869, 884, 888, 903, 918, 933, 934, 948, 949, 968, 983, 998, 1013, 1014

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			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. A003343, A345541, A345577, A345587, A345595, A345619, A345844.

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])

A345532 Numbers that are the sum of eight cubes in two or more ways.

Original entry on oeis.org

132, 139, 158, 160, 167, 174, 181, 186, 193, 195, 197, 200, 212, 216, 219, 223, 230, 237, 238, 244, 249, 251, 256, 258, 263, 265, 270, 272, 275, 277, 282, 284, 286, 288, 289, 291, 293, 296, 298, 300, 301, 303, 307, 308, 310, 312, 314, 315, 317, 319, 321, 322

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			139 is a term because 139 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 4^3 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 3^3.

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

Cf. A003331, A345489, A345520, A345533, A345541, A345577, A345784.

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, 8):
    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])

A345794 Numbers that are the sum of nine cubes in exactly two ways.

Original entry on oeis.org

72, 133, 140, 147, 159, 161, 166, 168, 175, 182, 185, 187, 189, 194, 196, 198, 201, 203, 205, 208, 213, 217, 220, 222, 227, 239, 243, 246, 252, 261, 265, 266, 273, 289, 296, 304, 306, 308, 323, 325, 328, 329, 330, 336, 342, 344, 349, 351, 352, 354, 356, 358

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345541 at term 25 because 224 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 6^3 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 3^3 + 4^3 + 4^3 + 4^3 = 1^3 + 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 4^3 + 5^3 = 2^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3.

Likely finite.

			133 is a term because 133 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 4^3 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 3^3.

Sean A. Irvine, Table of n, a(n) for n = 1..105

Cf. A345541, A345784, A345793, A345795, A345804, A345844.

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 == 2])
    for x in range(len(rets)):
        print(rets[x])

A345499 Numbers that are the sum of nine squares in two or more ways.

Original entry on oeis.org

24, 27, 30, 32, 33, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			27 is a term because 27 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 4^2 = 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A345489, A345498, A345500, A345509, A345541.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**2 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])

A345550 Numbers that are the sum of ten cubes in two or more ways.

Original entry on oeis.org

73, 80, 99, 134, 136, 141, 148, 155, 160, 162, 167, 169, 174, 176, 183, 186, 188, 190, 192, 193, 195, 197, 199, 202, 204, 206, 209, 211, 212, 213, 214, 216, 218, 221, 223, 225, 228, 230, 232, 235, 239, 240, 244, 246, 247, 249, 251, 253, 254, 258, 260, 262

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			80 is a term because 80 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 3^3 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3.

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

Cf. A003333, A345509, A345541, A345551, A345595, A345804.

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, 10):
    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])

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A345542 Numbers that are the sum of nine positive cubes in three or more ways.

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A345586 Numbers that are the sum of nine fourth powers in two or more ways.

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A345532 Numbers that are the sum of eight cubes in two or more ways.

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A345794 Numbers that are the sum of nine cubes in exactly two ways.

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A345499 Numbers that are the sum of nine squares in two or more ways.

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A345550 Numbers that are the sum of ten cubes in two or more ways.

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