A122730 Primes that are the sum of 6 positive cubes.

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

Author

Jonathan Vos Post, Sep 23 2006

Keywords

• easy
• nonn

Comments

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).

Examples

			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.
		

Links

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

Crossrefs

Cf. A000040, A003329.

Programs

Mathematica

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 *)

Formula

A000040 INTERSECTION A003329.

Extensions

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