A379400 -id:A379400 - cp's OEIS frontend

A379396 Numbers that can be written in exactly one way as a sum of at most nine positive third powers.

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 31, 38, 39, 45, 46, 47, 50, 52, 53

Offset: 1

Patrick De Geest, Dec 22 2024

The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes. (Cf. A002804)

			7 is in the sequence since there is only 1^3+1^3+1^3+1^3+1^3+1^3+1^3.
53 is in the sequence since there is only 1^3+1^3+2^3+2^3+2^3+3^3;

Eric Weisstein's World of Mathematics, Waring's Problem
Wikipedia, Waring's Problem
Index entries for sequences related to sums of cubes

Cf. A002804, A025453, A379397, A379398, A379399, A379400

upto(n) = my(v=vector(n), maxb=sqrtnint(n,3)); forvec(x=vector(9,i,[0,maxb]), s=sum(i=1,9,x[i]^3); if(0x==1,v,1) \\ David A. Corneth, Dec 23 2024

A379397 Numbers that can be written in exactly two different ways as a sum of at most nine positive third powers.

Original entry on oeis.org

8, 9, 16, 27, 28, 29, 30, 32, 33, 34, 36, 37, 40, 41, 42, 43, 44, 48, 49, 51, 54, 55, 57, 58, 59, 60, 61, 62, 63, 66, 69, 71, 73, 74, 76, 77, 78, 79, 80, 85, 87, 88, 90, 95, 101, 102, 103, 104, 106, 109, 111, 114, 115, 116, 117, 122, 123, 239

Offset: 1

Patrick De Geest, Dec 22 2024

The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes (cf. A002804).

			29 is in the sequence since 1^3 + 1^3 + 3^3 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 2^3 + 2^3.
123 is in the sequence since 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 3^3 + 3^3 + 4^3.

Wikipedia, Waring's Problem
Eric Weisstein's World of Mathematics, Waring's Problem
Index entries for sequences related to sums of cubes

Cf. A002804, A025453, A379396, A379398, A379399, A379400.

upto(n) = my(v=vector(n), maxb=sqrtnint(n,3)); forvec(x=vector(9,i,[0,maxb]), s=sum(i=1,9,x[i]^3); if(0x==2,v,1) \\ David A. Corneth, Dec 23 2024

A379398 Numbers that can be written in exactly three different ways as a sum of at most nine positive third powers.

Original entry on oeis.org

35, 56, 64, 65, 67, 68, 70, 75, 81, 82, 83, 84, 86, 89, 92, 93, 94, 96, 97, 98, 99, 100, 105, 107, 108, 110, 112, 113, 118, 119, 120, 121, 124, 125, 127, 130, 141, 142, 143, 148, 149, 150, 151, 167, 169, 174, 175, 176, 177, 178, 183, 186, 188, 202, 204, 212, 213, 214, 240, 247, 303

Offset: 1

Patrick De Geest, Dec 22 2024

nonn

The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes. (Cf. A002804)

			67 is in the sequence since 1^3+1^3+1^3+4^3 = 2^3+2^3+2^3+2^3+2^3+3^3 = 1^3+1^3+1^3+1^3+1^3+2^3+3^3+3^3.

Eric Weisstein's World of Mathematics, Waring's Problem
Wikipedia, Waring's Problem
Index entries for sequences related to sums of cubes

Cf. A002804, A025453, A379396, A379397, A379399, A379400

upto(n) = my(v=vector(n), maxb=sqrtnint(n,3)); forvec(x=vector(9,i,[0,maxb]), s=sum(i=1,9,x[i]^3); if(0x==3,v,1)) \\ David A. Corneth, Dec 23 2024

A379399 Numbers that can be written in exactly four different ways as a sum of at most nine positive third powers.

Original entry on oeis.org

72, 91, 126, 128, 129, 131, 132, 134, 135, 136, 137, 138, 139, 140, 144, 146, 147, 154, 155, 156, 157, 158, 162, 164, 165, 166, 168, 170, 171, 172, 173, 179, 180, 181, 184, 185, 187, 191, 193, 194, 195, 199, 203, 205, 206, 207, 210, 211, 215, 221, 228, 229, 230, 231, 232, 241, 242, 266, 267, 293, 295, 319, 330, 338, 366, 455

Offset: 1

Patrick De Geest, Dec 23 2024

nonn

The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes (cf. A002804).

Conjecture: this sequence is finite and a(66) = 455 is the last term. Verified up to 10^8. - Charles R Greathouse IV, Dec 28 2024

			215 is in the sequence since 1^3+2^3+3^3+3^3+3^3+5^3 = 1^3+1^3+2^3+2^3+2^3+4^3+5^3 = 2^3+2^3+3^3+3^3+3^3+3^3+3^3+4^3 = 1^3+2^3+2^3+2^3+2^3+3^3+3^3+4^3+4^3.

Eric Weisstein's World of Mathematics, Waring's Problem.
Wikipedia, Waring's Problem.
Index entries for sequences related to sums of cubes.

Cf. A002804, A025453, A379396, A379397, A379398, A379400.

upto(n) = my(v=vector(n), maxb=sqrtnint(n, 3)); forvec(x=vector(9, i, [0, maxb]), s=sum(i=1, 9, x[i]^3); if(0x==4, v, 1) \\ David A. Corneth, Dec 23 2024

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A379396 Numbers that can be written in exactly one way as a sum of at most nine positive third powers.

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A379397 Numbers that can be written in exactly two different ways as a sum of at most nine positive third powers.

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A379398 Numbers that can be written in exactly three different ways as a sum of at most nine positive third powers.

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A379399 Numbers that can be written in exactly four different ways as a sum of at most nine positive third powers.

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