A345833 -id:A345833 - cp's OEIS frontend

A003342 Numbers that are the sum of 8 positive 4th powers.

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, 263, 278, 293, 308, 323, 328, 338, 343, 353, 358, 368, 373, 388, 403, 408, 418, 423, 433, 438, 453, 468, 483, 488, 498, 503, 518, 533, 548, 563, 568

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 04 2020: (Start)
5396 is in the sequence as 5396 = 1^4 + 1^4 + 4^4 + 5^4 + 5^4 + 6^4 + 6^4 + 6^4.
8789 is in the sequence as 8789 = 5^4 + 5^4 + 5^4 + 5^4 + 6^4 + 6^4 + 6^4 + 7^4.
12469 is in the sequence as 12469 = 1^4 + 3^4 + 4^4 + 4^4 + 5^4 + 5^4 + 5^4 + 10^4. (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, A003331, A003341, A003343, A003353, A345577, A345833.

Select[Range[500], AnyTrue[PowersRepresentations[#, 8, 4], First[#]>0&]&] (* Jean-François Alcover, Jul 18 2017 *)

from itertools import combinations_with_replacement as mc
from sympy import integer_nthroot
def iroot4(n): return integer_nthroot(n, 4)[0]
def aupto(lim):
    pows4 = set(i**4 for i in range(1, iroot4(lim)+1) if i**4 <= lim)
    return sorted(t for t in set(sum(c) for c in mc(pows4, 8)) if t <= lim)
print(aupto(568)) # Michael S. Branicky, Aug 23 2021

A345834 Numbers that are the sum of eight fourth powers in exactly two ways.

Original entry on oeis.org

263, 278, 293, 308, 323, 343, 358, 373, 388, 423, 438, 453, 503, 533, 548, 563, 583, 598, 613, 628, 678, 693, 758, 773, 788, 803, 853, 868, 887, 902, 917, 932, 933, 967, 982, 997, 1028, 1043, 1047, 1062, 1108, 1127, 1142, 1157, 1172, 1222, 1237, 1283, 1302

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345577 at term 14 because 518 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 4^4 + 4^4 = 1^4 + 1^4 + 1^4 + 2^4 + 3^4 + 3^4 + 3^4 + 4^4 = 2^4 + 2^4 + 3^4 + 3^4 + 3^4 + 3^4 + 3^4 + 3^4.

			278 is a term because 278 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 4^4 = 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. A345577, A345784, A345824, A345833, A345835, A345844, A346327.

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

A345823 Numbers that are the sum of seven fourth powers in exactly one ways.

Original entry on oeis.org

7, 22, 37, 52, 67, 82, 87, 97, 102, 112, 117, 132, 147, 162, 167, 177, 182, 197, 212, 227, 242, 247, 322, 327, 337, 352, 387, 402, 407, 417, 452, 467, 482, 487, 562, 567, 577, 582, 592, 627, 631, 642, 646, 657, 661, 662, 676, 691, 692, 706, 707, 711, 721, 722

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A003341 at term 23 because 262 = 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 + 4^4.

			22 is a term because 22 = 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. A003341, A345773, A345813, A345824, A345833, A346278.

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

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])

A346326 Numbers that are the sum of eight fifth powers in exactly one way.

Original entry on oeis.org

8, 39, 70, 101, 132, 163, 194, 225, 250, 256, 281, 312, 343, 374, 405, 436, 467, 492, 523, 554, 585, 616, 647, 678, 734, 765, 796, 827, 858, 889, 976, 1007, 1031, 1038, 1062, 1069, 1093, 1100, 1124, 1155, 1186, 1217, 1218, 1248, 1249, 1273, 1280, 1304, 1311

Offset: 1

David Consiglio, Jr., Jul 13 2021

nonn

Differs from A003353 at term 156 because 4100 = 1^5 + 1^5 + 1^5 + 3^5 + 3^5 + 3^5 + 3^5 + 5^5 = 1^5 + 1^5 + 1^5 + 1^5 + 4^5 + 4^5 + 4^5 + 4^5.

			8 is a term because 8 = 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. A003353, A345833, A346278, A346327, A346336.

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, 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])

A345783 Numbers that are the sum of eight cubes in exactly one way.

Original entry on oeis.org

8, 15, 22, 29, 34, 36, 41, 43, 48, 50, 55, 57, 60, 62, 64, 67, 69, 71, 74, 76, 78, 81, 83, 85, 86, 88, 92, 93, 95, 97, 99, 100, 102, 104, 106, 107, 111, 112, 113, 114, 118, 119, 120, 121, 123, 125, 126, 130, 133, 134, 137, 138, 140, 141, 144, 145, 146, 148

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A003331 at term 49 because 132 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 5^3 = 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3.

Likely finite.

			15 is a term because 15 = 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..209

Cf. A003331, A345773, A345784, A345793, A345833.

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

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A003342 Numbers that are the sum of 8 positive 4th powers.

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A345834 Numbers that are the sum of eight fourth powers in exactly two ways.

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A345823 Numbers that are the sum of seven fourth powers in exactly one ways.

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

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A346326 Numbers that are the sum of eight fifth powers in exactly one way.

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A345783 Numbers that are the sum of eight cubes in exactly one way.

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