A003330 -id:A003330 - cp's OEIS frontend

A004829 Numbers that are the sum of at most 7 positive cubes.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

N. J. A. Sloane

McCurley proves that every n > exp(exp(13.97)) is in A003330 and hence in this sequence. Siksek proves that all n > 454 are in this sequence. - Charles R Greathouse IV, Jun 29 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Jan Bohman and Carl-Erik Froberg, Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
K. S. McCurley, An effective seven-cube theorem, J. Number Theory, 19 (1984), 176-183.
Samir Siksek, Every integer greater than 454 is the sum of at most seven positive cubes, Algebra and Number Theory 10:10 (2016), pp. 2093-2119.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences related to sums of cubes

Complement of A018889; subsequence of A003330.
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Cf. A018888.

A345520 Numbers that are the sum of seven cubes in two or more ways.

Original entry on oeis.org

131, 159, 166, 173, 185, 192, 211, 222, 229, 236, 243, 248, 255, 257, 262, 264, 269, 274, 276, 281, 283, 285, 288, 290, 292, 295, 299, 300, 302, 307, 309, 311, 314, 318, 320, 321, 325, 332, 333, 337, 339, 340, 344, 346, 348, 351, 353, 355, 358, 359, 360, 363

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			159 is a term because 159 = 1^3 + 1^3 + 1^3 + 1^3 + 3^3 + 3^3 + 3^3 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 4^3.

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

Cf. A003330, A345479, A345511, A345521, A345532, A345568, A345774.

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, 7):
    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])

A332107 Numbers that are not the sum of seven (7) positive cubes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 36, 37, 38, 39, 41, 43, 44, 45, 46, 48, 50, 51, 52, 53, 55, 57, 58, 60, 62, 63, 64, 65, 67, 69, 71, 72, 74, 76, 78, 79, 81, 82, 83, 86, 88, 89, 90, 93, 95, 97, 100

Offset: 1

M. F. Hasler, Aug 24 2020

The sequence is finite, with last term a(208) = 2408.

			The smallest positive numbers not in the sequence are 7 = 7 * 1^3, 14 = 2^3 + 6 * 1^3, 21 = 2 * 2^3 + 5 * 1^3, ...
The last 10 terms of the sequence are a(199 .. 208) = {1078, 1094, 1364, 1409, 1579, 1582, 1796, 2030, 2382, 2408}.

M. F. Hasler, Table of n, a(n) for n = 1..208 (full sequence).
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024. See p. 4.

Complement of A003330.

Cf. A332108, A332109, A332110, A332111: analog for eight, ..., eleven cubes.

Select[Range[100], (pr = PowersRepresentations[#, 7, 3][[;; , 1]]) == {} || Max[pr] == 0 &] (* Amiram Eldar, Aug 24 2020 *)

A332107=setminus([1..2440],A003330_upto(2444))

A003386 Numbers that are the sum of 8 nonzero 8th powers.

Original entry on oeis.org

8, 263, 518, 773, 1028, 1283, 1538, 1793, 2048, 6568, 6823, 7078, 7333, 7588, 7843, 8098, 8353, 13128, 13383, 13638, 13893, 14148, 14403, 14658, 19688, 19943, 20198, 20453, 20708, 20963, 26248, 26503, 26758, 27013, 27268, 32808, 33063, 33318, 33573

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
9534597 is in the sequence as 9534597 = 2^8 + 3^8 + 3^8 + 3^8 + 5^8 + 6^8 + 6^8 + 7^8.
13209988 is in the sequence as 13209988 = 1^8 + 1^8 + 2^8 + 2^8 + 2^8 + 6^8 + 7^8 + 7^8.
19046628 is in the sequence as 19046628 = 2^8 + 2^8 + 3^8 + 4^8 + 6^8 + 7^8 + 7^8 + 7^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

M = 92646056; m = M^(1/8) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++,
s = a^8 + b^8 + c^8 + d^8 + e^8 + f^8 + g^8 + h^8;
If[s <= M, Sow[s]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

b-file checked by R. J. Mathar, Aug 01 2020

Incorrect program removed by David A. Corneth, Aug 01 2020

A345478 Numbers that are the sum of seven squares in one or more ways.

Original entry on oeis.org

7, 10, 13, 15, 16, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 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

Offset: 1

David Consiglio, Jr., Jun 19 2021

nonn

			10 is a term because 10 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2.

Sean A. Irvine, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A003330, A344805, A345479, A345488, A345508.

ssQ[n_]:=Count[IntegerPartitions[n,{7}],?(AllTrue[Sqrt[#],IntegerQ]&)]>0; Select[ Range[ 80],ssQ] (* _Harvey P. Dale, Jun 22 2022 *)

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, 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])

From Chai Wah Wu, Jun 12 2025: (Start)
All integers >= 21 are terms. See A345508 for a similar proof.
a(n) = 2*a(n-1) - a(n-2) for n > 9.
G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 4*x + 7)/(x - 1)^2. (End)

A345773 Numbers that are the sum of seven cubes in exactly one way.

Original entry on oeis.org

7, 14, 21, 28, 33, 35, 40, 42, 47, 49, 54, 56, 59, 61, 66, 68, 70, 73, 75, 77, 80, 84, 85, 87, 91, 92, 94, 96, 98, 99, 103, 105, 106, 110, 111, 112, 113, 117, 118, 122, 124, 125, 129, 132, 133, 136, 137, 138, 140, 143, 144, 145, 147, 148, 150, 151, 152, 154

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A003330 at term 44 because 131 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 5^3 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3.

Likely finite.

			14 is a term because 14 = 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..324

Cf. A003330, A048929, A345774, A345783, A345823.

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, 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])

A122731 Primes that are the sum of 7 positive cubes.

Original entry on oeis.org

7, 47, 59, 61, 73, 103, 113, 131, 137, 151, 157, 163, 173, 181, 197, 199, 211, 223, 227, 229, 241, 257, 263, 269, 271, 281, 283, 307, 311, 313, 337, 347, 349, 353, 359, 367, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 449, 457, 461, 463, 467, 479, 487

Offset: 1

Jonathan Vos Post, Sep 23 2006

By parity, there must be an odd number of odds in the sum. Hence this sequence is the union of primes which are the sum of seven odd cubes (such as 7 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3); primes which are the sum of an two even and five odd cubes (such as 229 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 6^3); primes which are the sum of the cube of four even numbers and the cubes of three odd numbers (such as 61 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3); and the primes which are the sum of the cube of an odd number and the cubes of six even numbers (such as 173 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 5^3). A subset of this sequence is the primes which are the sum of the cubes of seven distinct primes (i.e. of the form p^3 + q^3 + r^3 + s^3 + t^3 + u^3 + v^3 for p, q, r, s, t, u, v distinct odd primes) such as 112759 = 3^3 + 5^3 + 7^3 + 11^3 + 13^3 + 17^3 + 47^3. Another subsequence is the primes which are the sum of seven cubes in two different ways, or three different ways. No prime can be the sum of two cubes (by factorization of the sum of two cubes).

			a(1) = 7 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3.
a(4) = 61 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A003330.

nn=500; lim = Floor[(nn-6)^(1/3)]; Select[Union[Total /@ Tuples[Range[lim]^3, {7}]], # <= nn && PrimeQ[#] &]  (* Harvey P. Dale, Mar 13 2011 *)

A000040 INTERSECTION A003330.

More terms from R. J. Mathar, Jun 13 2007

cp's OEIS Frontend

A004829 Numbers that are the sum of at most 7 positive cubes.

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A345520 Numbers that are the sum of seven cubes in two or more ways.

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A332107 Numbers that are not the sum of seven (7) positive cubes.

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A003386 Numbers that are the sum of 8 nonzero 8th powers.

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A345478 Numbers that are the sum of seven squares in one or more ways.

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A345773 Numbers that are the sum of seven cubes in exactly one way.

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A122731 Primes that are the sum of 7 positive cubes.

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