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

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

Author

Patrick De Geest, Dec 23 2024

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

PARI

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