A345853 -id:A345853 - cp's OEIS frontend

A003344 Numbers that are the sum of 10 positive 4th powers.

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, 265, 275, 280, 290, 295, 310, 325, 330, 340, 345, 355, 360, 370, 375, 385, 390, 400, 405, 410, 420, 425, 435, 440, 450, 455, 465

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
5176 is in the sequence as 5176 = 2^4 + 2^4 + 3^4 + 3^4 + 3^4 + 5^4 + 5^4 + 5^4 + 5^4 + 7^4.
6901 is in the sequence as 6901 = 1^4 + 4^4 + 4^4 + 5^4 + 5^4 + 5^4 + 5^4 + 6^4 + 6^4 + 6^4.
8502 is in the sequence as 8502 = 1^4 + 3^4 + 4^4 + 5^4 + 5^4 + 5^4 + 6^4 + 6^4 + 6^4 + 7^4. (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. A000583, A003333, A003343, A003355, A047721, A345595, A345853.

from itertools import count, takewhile, combinations_with_replacement as mc
def aupto(limit):
    pows4 = list(takewhile(lambda x: x <= limit, (i**4 for i in count(1))))
    sum10 = set(sum(c) for c in mc(pows4, 10) if sum(c) <= limit)
    return sorted(sum10)
print(aupto(465)) # Michael S. Branicky, Oct 25 2021

A345843 Numbers that are the sum of nine fourth powers in exactly one ways.

Original entry on oeis.org

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, 274, 329, 354, 369, 384, 409, 419, 434, 449, 484, 489, 499, 514, 569, 594, 609, 624, 633, 648, 649, 659, 663, 674, 678, 689, 693, 708

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A003343 at term 28 because 264 = 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 + 4^4.

			24 is a term because 24 = 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. A003343, A345793, A345833, A345844, A345853, A346336.

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 == 1])
    for x in range(len(rets)):
        print(rets[x])

A345854 Numbers that are the sum of ten fourth powers in exactly two ways.

Original entry on oeis.org

265, 280, 295, 310, 325, 340, 345, 355, 360, 375, 390, 405, 420, 425, 440, 455, 470, 485, 505, 565, 580, 585, 595, 630, 645, 660, 665, 695, 710, 725, 745, 760, 805, 820, 835, 840, 870, 885, 889, 900, 904, 919, 920, 934, 935, 949, 950, 964, 965, 969, 984, 999

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345595 at term 20 because 520 = 1^4 + 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 + 1^4 + 2^4 + 3^4 + 3^4 + 3^4 + 4^4 = 1^4 + 1^4 + 2^4 + 2^4 + 3^4 + 3^4 + 3^4 + 3^4 + 3^4 + 3^4.

			280 is a term because 280 = 1^4 + 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 + 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. A345595, A345804, A345844, A345853, A345855, A346347.

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

A346346 Numbers that are the sum of ten fifth powers in exactly one way.

Original entry on oeis.org

10, 41, 72, 103, 134, 165, 196, 227, 252, 258, 283, 289, 314, 320, 345, 376, 407, 438, 469, 494, 500, 525, 531, 556, 587, 618, 649, 680, 711, 736, 742, 767, 798, 829, 860, 891, 922, 953, 978, 1009, 1033, 1040, 1064, 1071, 1095, 1102, 1126, 1133, 1157, 1164

Offset: 1

David Consiglio, Jr., Jul 13 2021

nonn

Differs from A003355 at term 229 because 4102 = 1^5 + 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 + 1^5 + 4^5 + 4^5 + 4^5 + 4^5.

			10 is a term because 10 = 1^5 + 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. A003355, A345853, A346336, A346347.

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, 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])

A345803 Numbers that are the sum of ten cubes in exactly one ways.

Original entry on oeis.org

10, 17, 24, 31, 36, 38, 43, 45, 50, 52, 57, 59, 62, 64, 66, 69, 71, 76, 78, 83, 85, 87, 88, 90, 92, 94, 95, 97, 101, 102, 104, 106, 108, 109, 111, 113, 114, 115, 116, 118, 120, 121, 122, 123, 125, 127, 128, 129, 130, 132, 135, 137, 139, 140, 142, 143, 146

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A003333 at term 18 because 73 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 4^3 = 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3.

Likely finite.

			17 is a term because 17 = 1^3 + 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..151

Cf. A003333, A345793, A345804, A345853.

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 == 1])
    for x in range(len(rets)):
        print(rets[x])

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A003344 Numbers that are the sum of 10 positive 4th powers.

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A345843 Numbers that are the sum of nine fourth powers in exactly one ways.

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A345854 Numbers that are the sum of ten fourth powers in exactly two ways.

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A346346 Numbers that are the sum of ten fifth powers in exactly one way.

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A345803 Numbers that are the sum of ten cubes in exactly one ways.

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