A332107 -id:A332107 - cp's OEIS frontend

A003334 Numbers that are the sum of 11 positive cubes.

11, 18, 25, 32, 37, 39, 44, 46, 51, 53, 58, 60, 63, 65, 67, 70, 72, 74, 77, 79, 81, 84, 86, 88, 89, 91, 93, 95, 96, 98, 100, 102, 103, 105, 107, 109, 110, 112, 114, 115, 116, 117, 119, 121, 122, 123, 124, 126, 128, 129, 130, 131, 133, 135, 136, 137, 138, 140, 141, 142, 143, 144

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

The sequence contains all integers greater than 321 which is the last of only 92 positive integers not in this sequence. - M. F. Hasler, Aug 25 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
1120 is in the sequence as 1120 = 2^3 + 3^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 5^3 +  8^3.
2339 is in the sequence as 2339 = 4^3 + 4^3 + 4^3 + 4^3 + 5^3 + 5^3 + 5^3 + 5^3 + 5^3 + 9^3 +  9^3.
3594 is in the sequence as 3594 = 4^3 + 5^3 + 6^3 + 6^3 + 6^3 + 6^3 + 7^3 + 7^3 + 7^3 + 8^3 + 10^3. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Other sequences S(k, m) of numbers that are the sum of k nonzero m-th powers:
  A000404 = S(2, 2), A000408 = S(3, 2), A000414 = S(4, 2) complement of A000534,
  A047700 = S(5, 2) complement of A047701, A180968 = complement of S(6,2);
  A003325 = S(2, 3), A003072 = S(3, 3), A003327 .. A003335 = S(4 .. 12, 3) and A332107 .. A332111 = complement of S(7 .. 11, 3);
  A003336 .. A003346 = S(2 .. 12, 4), A003347 .. A003357 = S(2 .. 12, 5),
  A003358 .. A003368 = S(2 .. 12, 6), A003369 .. A003379 = S(2 .. 12, 7),
  A003380 .. A003390 = S(2 .. 12, 8), A003391 .. A004801 = S(2 .. 12, 9),
  A004802 .. A004812 = S(2 .. 12, 10), A004813 .. A004823 = S(2 .. 12, 11).

(A003334_upto(N, k=11, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(150) \\ See also A003333 for alternate code. - M. F. Hasler, Aug 03 2020

a(n) = n + 92 for all n > 229. - M. F. Hasler, Aug 25 2020

A332108 Numbers that are not the sum of eight (8) positive cubes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 37, 38, 39, 40, 42, 44, 45, 46, 47, 49, 51, 52, 53, 54, 56, 58, 59, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 79, 80, 82, 84, 87, 89, 90, 91, 94, 96, 98

Offset: 1

M. F. Hasler, Aug 24 2020

The sequence is finite, with last term a(142) = 620.

			The smallest positive numbers not in the sequence are:
  8 = 8 * 1^3, 15 = 2^3 + 7 * 1^3, 22 = 2 * 2^3 + 6 * 1^3,
  29 = 3 * 2^3 + 5 * 1^3 and then 34 = 3^3 + 7 * 1^3.
The last 10 terms of the sequence are a(133 .. 142) = {372, 381, 395, 407, 414, 421, 444, 463, 470, 620}.

M. F. Hasler, Table of n, a(n) for n = 1..142
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 A003331.

Cf. A332107, A332109, A332110 (analog for 7, 9 resp. 10 cubes).

Select[Range[650], (pr = PowersRepresentations[#, 8, 3][[;; , 1]]) == {} || Max[pr] == 0 &] (* Amiram Eldar, Aug 25 2020 *)

A332108=setminus([1..620],A003331_upto(620))

A332109 Numbers that are not the sum of nine (9) positive cubes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 36, 38, 39, 40, 41, 43, 45, 46, 47, 48, 50, 52, 53, 54, 55, 57, 59, 60, 62, 64, 66, 67, 69, 71, 73, 74, 76, 78, 80, 81, 83, 85, 88, 90, 92, 95, 97, 99, 102

Offset: 1

M. F. Hasler, Aug 24 2020

The sequence is finite, with last term a(114) = 471.

			The smallest positive numbers not in the sequence are:
  9 = 9 * 1^3, 16 = 2^3 + 8 * 1^3, 23 = 2 * 2^3 + 7 * 1^3,
  30 = 3 * 2^3 + 6 * 1^3 and then 35 = 3^3 + 8 * 1^3.
The last 10 terms of the sequence are a(105 .. 114) = {293, 305, 310, 312, 319, 347, 366, 373, 422, 471}.

M. F. Hasler, Table of n, a(n) for n = 1..114 (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 A003332.

Cf. A332107, A332108, A332110 (analog for 7, 8 and 10 cubes, respectively).

Select[Range[500], (pr = PowersRepresentations[#, 9, 3][[;; , 1]]) == {} || Max[pr] == 0 &] (* Amiram Eldar, Aug 24 2020 *)

A332109=setminus([1..555],A003332_upto(666))

A332110 Numbers that are not the sum of ten (10) positive cubes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 37, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 53, 54, 55, 56, 58, 60, 61, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 89, 91, 93, 96, 98, 100

Offset: 1

M. F. Hasler, Aug 24 2020

The sequence is finite, with last term a(99) = 374.

			The smallest positive numbers not in the sequence are:
  10 = 10 * 1^3, 17 = 2^3 + 9 * 1^3, 24 = 2 * 2^3 + 8 * 1^3,
  31 = 3 * 2^3 + 7 * 1^3 and then 36 = 3^3 + 9 * 1^3.
The last 10 terms of the sequence are a(90 .. 99) = {196, 201, 208, 215, 222, 257, 294, 313, 320, 374}.

M. F. Hasler, Table of n, a(n) for n = 1..99 (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 A003333.

Cf. A332107, A332108, A332109, A332111 (analog for 7, 8, 9, resp. 11 cubes).

Select[Range[400], (pr = PowersRepresentations[#, 10, 3][[;; , 1]]) == {} || Max[pr] == 0 &] (* Amiram Eldar, Aug 24 2020 *)

A332110=setminus([1..444],A003333_upto(555))

A332111 Numbers that are not the sum of eleven (11) positive cubes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 40, 41, 42, 43, 45, 47, 48, 49, 50, 52, 54, 55, 56, 57, 59, 61, 62, 64, 66, 68, 69, 71, 73, 75, 76, 78, 80, 82, 83, 85, 87, 90, 92, 94, 97, 99

Offset: 1

M. F. Hasler, Aug 25 2020

The sequence is finite, with last term a(92) = 321.

			The smallest positive numbers not in the sequence are:
  11 = 11 * 1^3, 18 = 2^3 + 10 * 1^3, 25 = 2 * 2^3 + 9 * 1^3,
  32 = 3 * 2^3 + 8 * 1^3 and then 37 = 3^3 + 10 * 1^3.
The last 23 terms of the sequence (not in the data section) are a(70 .. 92) = {101, 104, 106, 108, 111, 113, 118, 120, 125, 127, 132, 134, 139, 146, 153, 160, 171, 190, 197, 209, 216, 223, 321}.

M. F. Hasler, Table of n, a(n) for n = 1..92 (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 A003334.

Cf. A332107, A332108, A332109, A332110 (analog for 7, 8, 9 and 10 cubes).

Select[Range[400], (pr = PowersRepresentations[#, 11, 3][[;; , 1]]) == {} || Max[pr] == 0 &] (* adapted from Amiram Eldar's code for A332110 *)

A332111=setminus([1..333],A003333_upto(444))

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A003334 Numbers that are the sum of 11 positive cubes.

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A332108 Numbers that are not the sum of eight (8) positive cubes.

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A332109 Numbers that are not the sum of nine (9) positive cubes.

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A332110 Numbers that are not the sum of ten (10) positive cubes.

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A332111 Numbers that are not the sum of eleven (11) positive cubes.

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