A003329 -id:A003329 - cp's OEIS frontend

A048930 Numbers that are the sum of 6 positive cubes in exactly 2 ways.

158, 165, 184, 228, 235, 247, 256, 261, 268, 273, 275, 280, 282, 284, 287, 291, 294, 306, 310, 313, 317, 324, 331, 332, 343, 345, 347, 350, 352, 362, 371, 373, 376, 378, 380, 385, 387, 388, 392, 395, 399, 404, 406, 408, 418, 425, 430, 432, 436, 437, 441

Offset: 1

Eric W. Weisstein

It appears that this sequence has 1094 terms, the last of which is 21722. - Donovan Johnson, Jan 09 2013

			158 is in the sequence since 158 = 64+64+27+1+1+1 = 125+8+8+8+8+1.

Donovan Johnson, Table of n, a(n) for n = 1..1094 (terms < 10^7)
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003329, A048927, A048929, A048931, A345511, A345774, A345814.

Reap[For[n = 1, n <= 500, n++, pr = Select[ PowersRepresentations[n, 6, 3], Times @@ # != 0 &]; If[pr != {} && Length[pr] == 2, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)

mx=10^6; ct=vector(mx); cb=vector(99); for(i=1, 99, cb[i]=i^3); for(i1=1, 99, s1=cb[i1]; for(i2=i1, 99, s2=s1+cb[i2]; if(s2+4*cb[i2]>mx, next(2)); for(i3=i2, 99, s3=s2+cb[i3]; if(s3+3*cb[i3]>mx, next(2)); for(i4=i3, 99, s4=s3+cb[i4]; if(s4+2*cb[i4]>mx, next(2)); for(i5=i4, 99, s5=s4+cb[i5]; if(s5+cb[i5]>mx, next(2)); for(i6=i5, 99, s6=s5+cb[i6]; if(s6>mx, next(2)); ct[s6]++)))))); n=0; for(i=6, mx, if(ct[i]==2, n++; write("b048930.txt", n " " i))) /* Donovan Johnson, Jan 09 2013 */

Terms corrected by Donovan Johnson, Jan 09 2013

A048931 Numbers that are the sum of 6 positive cubes in exactly 3 ways.

Original entry on oeis.org

221, 254, 369, 411, 443, 469, 495, 502, 576, 595, 600, 648, 658, 684, 704, 711, 720, 739, 746, 753, 760, 765, 767, 772, 774, 779, 786, 793, 811, 818, 828, 835, 844, 854, 863, 866, 874, 880, 884, 886, 892, 893, 899, 905, 910, 919, 928, 929, 935, 936, 937

Offset: 1

Eric W. Weisstein

It appears that this sequence has 1141 terms, the last of which is 26132. - Donovan Johnson, Jan 09 2013

			221 is in the sequence since 221 = 216+1+1+1+1+1 = 125+64+8+8+8+8 = 64+64+64+27+1+1.

Donovan Johnson, Table of n, a(n) for n = 1..1141 (terms < 10^7)
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003329, A048929, A048930, A343705, A345512, A345766, A345775, A345815.

Reap[For[n = 1, n <= 1000, n++, pr = Select[ PowersRepresentations[n, 6, 3], Times @@ # != 0 &]; If[pr != {} && Length[pr] == 3, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)

mx=10^6; ct=vector(mx); cb=vector(99); for(i=1, 99, cb[i]=i^3); for(i1=1, 99, s1=cb[i1]; for(i2=i1, 99, s2=s1+cb[i2]; if(s2+4*cb[i2]>mx, next(2)); for(i3=i2, 99, s3=s2+cb[i3]; if(s3+3*cb[i3]>mx, next(2)); for(i4=i3, 99, s4=s3+cb[i4]; if(s4+2*cb[i4]>mx, next(2)); for(i5=i4, 99, s5=s4+cb[i5]; if(s5+cb[i5]>mx, next(2)); for(i6=i5, 99, s6=s5+cb[i6]; if(s6>mx, next(2)); ct[s6]++)))))); n=0; for(i=6, mx, if(ct[i]==3, n++; write("b048931.txt", n " " i))) /* Donovan Johnson, Jan 09 2013 */

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Oct 02 2000

A048929 Numbers that are the sum of 6 positive cubes in exactly 1 way.

Original entry on oeis.org

6, 13, 20, 27, 32, 34, 39, 41, 46, 48, 53, 58, 60, 65, 67, 69, 72, 76, 79, 83, 84, 86, 90, 91, 95, 97, 98, 102, 104, 105, 109, 110, 116, 117, 121, 123, 124, 128, 130, 132, 135, 136, 137, 139, 142, 143, 144, 146, 147, 151, 153, 154, 156, 160, 161, 162, 163, 170

Offset: 1

Eric W. Weisstein

It appears that this sequence has 841 terms, the last of which is 19417. This means that all numbers greater than 19417 can be written as the sum of six positive cubes in at least two ways. - T. D. Noe, Dec 13 2006

T. D. Noe, Table of n, a(n) for n=1..841
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003329, A048930, A048931.

Cf. A057907 (numbers not the sum of six positive cubes)

Select[ Range[200], Length[ Select[ PowersRepresentations[#, 6, 3], And @@ (Positive /@ #) &]] == 1 &] (* Jean-François Alcover, Oct 25 2012 *)

from collections import Counter
from itertools import combinations_with_replacement as multi_combs
def aupto(lim):
  c = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2)))
  s = filter(lambda x: x<=lim, (sum(mc) for mc in multi_combs(c, 6)))
  counts = Counter(s)
  return sorted(k for k in counts if counts[k]==1)
print(aupto(20000)) # Michael S. Branicky, Jun 13 2021

A345511 Numbers that are the sum of six cubes in two or more ways.

Original entry on oeis.org

158, 165, 184, 221, 228, 235, 247, 254, 256, 261, 268, 273, 275, 280, 282, 284, 287, 291, 294, 306, 310, 313, 317, 324, 331, 332, 343, 345, 347, 350, 352, 362, 369, 371, 373, 376, 378, 380, 385, 387, 388, 392, 395, 399, 404, 406, 408, 411, 418, 425, 430, 432

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

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

Cf. A003329, A048930, A343702, A344806, A345512, A345520, A345559.

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, 6):
    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])

A057907 Positive integers that are not the sum of precisely six positive cubes.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 35, 36, 37, 38, 40, 42, 43, 44, 45, 47, 49, 50, 51, 52, 54, 55, 56, 57, 59, 61, 62, 63, 64, 66, 68, 70, 71, 73, 74, 75, 77, 78, 80, 81, 82, 85, 87, 88, 89, 92, 93, 94, 96

Offset: 1

Eric W. Weisstein

It appears that this sequence has 492 terms, the last of which is 19202. - T. D. Noe, Dec 13 2006

T. D. Noe, Table of n, a(n) for n=1..492
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024. p. 4.
Eric Weisstein's World of Mathematics, Cubic Number.

Numbers not in (complement of) A003329.

Cf. A048929 (numbers that are the sum of six positive cubes in exactly 1 way).

Select[ Range[100], Length[ Select[ PowersRepresentations[#, 6, 3], And @@ (Positive /@ #) &]] != 1 &] (* Jean-François Alcover, Oct 25 2012 *)

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

A344805 Numbers that are the sum of six squares in one or more ways.

Original entry on oeis.org

6, 9, 12, 14, 15, 17, 18, 20, 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

Offset: 1

David Consiglio, Jr., Jun 19 2021

nonn

			9 is a term because 9 = 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. A003329, A047700, A344806, A345478, A345508.

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, 6):
    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 >= 20 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 - 3*x + 6)/(x - 1)^2. (End)

A025459 Number of partitions of n into 6 positive cubes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0

Offset: 0

David W. Wilson

nonn

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of cubes

Cf. A003329, A048929, A048930, A048931.

A025459 := proc(n)
    local a,x,y,z,u,v,wcu ;
    a := 0 ;
    for x from 1 do
        if 6*x^3 > n then
            return a;
        end if;
        for y from x do
            if x^3+5*y^3 > n then
                break;
            end if;
            for z from y do
                if x^3+y^3+4*z^3 > n then
                    break;
                end if;
                for u from z do
                    if x^3+y^3+z^3+3*u^3 > n then
                        break;
                    end if;
                    for v from u do
                        if x^3+y^3+z^3+u^3+2*v^3 > n then
                            break;
                        end if;
                        wcu := n-x^3-y^3-z^3-u^3-v^3 ;
                        if isA000578(wcu) then
                            a := a+1 ;
                        end if;
                    end do:
                end do:
            end do:
        end do:
    end do:
end proc: # R. J. Mathar, Sep 15 2015
# Alternative:
N:= 200:
G:= mul(1/(1-y*x^(k^3)),k=1..floor(N^(1/3))):
C6:= coeff(series(G,y,7),y,6):
S:= series(C6,x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, May 10 2020

a[n_] := Count[PowersRepresentations[n, 6, 3], pr_List /; FreeQ[pr, 0]];
a /@ Range[0, 200] (* Jean-François Alcover, Jun 22 2020 *)
Table[Count[IntegerPartitions[n,{6}],?(AllTrue[Surd[#,3],IntegerQ]&)],{n,0,110}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jun 06 2021 *)

a(n) = [x^n y^6] Product_{k>=1} 1/(1 - y*x^(k^3)). - Ilya Gutkovskiy, Apr 23 2019

A069137 Numbers which are sums of neither 1, 2, 3, 4, 5 or 6 nonnegative cubes.

Original entry on oeis.org

7, 14, 15, 21, 22, 23, 42, 47, 49, 50, 61, 77, 85, 87, 103, 106, 111, 112, 113, 114, 122, 140, 148, 159, 166, 167, 174, 175, 178, 185, 186, 204, 211, 212, 223, 229, 230, 231, 237, 238, 239, 276, 292, 295, 300, 302, 303, 311, 327, 329, 337, 340, 356, 363, 364

Offset: 1

N. J. A. Sloane, Apr 08 2002; edited Sep 15 2006

nonn

Sequence is conjectured to be finite.

			Numbers which need at least seven terms to represent them as a sum of positive cubes: 14=8+1+1+1+1+1+1.

Bohman, Jan and Froberg, Carl-Erik; Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
F. Romani, Computations concerning Waring's problem, Calcolo, 19 (1982), 415-431.

T. D. Noe, Table of n, a(n) for n=1..138
Jean-Marc Deshouillers, Francois Hennecart and Bernard Landreau; appendix by I. Gusti Putu Purnaba, 7373170279850, Math. Comp. 69 (2000), 421-439.
Index entries for sequences related to sums of cubes

Cf. A057907, A003329, A003325, A003072, A003327, A003328, A000578, A085334.

Natural numbers remaining if union of A003325, A003072, A003327, A003328, A003329 and A000578 sets were deleted. Remark: this sequence itself does not include cubes, in contrast to A085334.

A122730 Primes that are the sum of 6 positive cubes.

Original entry on oeis.org

13, 41, 53, 67, 79, 83, 97, 109, 137, 139, 151, 163, 173, 179, 191, 193, 199, 233, 241, 263, 271, 277, 313, 317, 331, 347, 359, 373, 383, 389, 397, 421, 433, 439, 443, 449, 457, 463, 467, 479, 499, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593

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 an even and five odd cubes (such as x1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3); primes which are the sum of the cube of three even numbers and the cubes of three odd numbers (such as 53 = 1^3 + 1^3 + 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 five even numbers (such as 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 or 67 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3). A subset of this sequence is the primes which are the sum of the cubes of six distinct primes (i.e. of the form 2^3 + p^3 + q^3 + r^3 + s^3 + t^3 for p, q, r, s, t distinct odd primes) such as 8693 = 2^3 + 3^3 + 5^3 + 7^3 + 11^3 + 19^3. Another subsequence is the primes which are the sum of six cubes in two different ways, such as 313 = 1^3 + 2^3 + 3^3 + 3^3 + 5^3 + 5^3 = 2^3 + 2^3 + 3^3 + 3^3 + 3^3 + 6^3. Similarly, another subsequence is the primes which are the sum of six cubes in three different ways, such as 443. No prime can be the sum of two cubes (by factorization of the sum of two cubes).

			a(1) = 13 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3.
a(2) = 41 = 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3
a(3) = 53 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 3^3.
a(4) = 67 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3.

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

Cf. A000040, A003329.

up = 900; q = Range[up^(1/3)]^3; a = {0}; Do[ b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 6}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 13 2016 *)

A000040 INTERSECTION A003329.

a(13)-a(53) from Giovanni Resta, Jun 13 2016

cp's OEIS Frontend

A048930 Numbers that are the sum of 6 positive cubes in exactly 2 ways.

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A048931 Numbers that are the sum of 6 positive cubes in exactly 3 ways.

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A048929 Numbers that are the sum of 6 positive cubes in exactly 1 way.

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A345511 Numbers that are the sum of six cubes in two or more ways.

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A057907 Positive integers that are not the sum of precisely six positive cubes.

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

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A344805 Numbers that are the sum of six squares in one or more ways.

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A025459 Number of partitions of n into 6 positive cubes.

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