A122728 Primes that are the sum of 4 positive cubes.

11, 37, 67, 89, 107, 137, 149, 163, 191, 193, 233, 271, 317, 353, 367, 379, 383, 409, 439, 461, 467, 479, 503, 523, 541, 587, 593, 601, 613, 631, 641, 653, 691, 709, 739, 751, 773, 809, 821, 839, 857, 863, 883, 887, 919, 929, 947, 971, 983, 991, 1033, 1069

Offset: 1

Author

Jonathan Vos Post, Sep 23 2006

Keywords

• easy
• nonn
• changed

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 the cube of an even number and the cubes of three odd numbers (such as 11 = 1^3 + 1^3 + 1^3 + 2^3) and the primes which are the sum of the cube of an odd number and the cubes of three even numbers (such as 149 = 2^3 + 2^3 + 2^3 + 5^3). A subset of this sequence is the primes which are the sum of the cubes of four distinct primes (i.e. of the form 2^3 + p^3 + q^3 + r^3 for p, q, r, distinct odd primes) such as 503 = 2^3 + 3^3 + 5^3 + 7^3; or 2357 = 2^3 + 3^3 + 5^3 + 13^3. No prime can be the sum of two cubes (by factorization of the sum of two cubes).

Examples

			a(1) = 11 = 1^3 + 1^3 + 1^3 + 2^3.
a(2) = 37 = 1^3 + 1^3 + 2^3 + 3^3.
a(3) = 67 = 1^3 + 1^3 + 1^3 + 4^3.
		

Links

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

Crossrefs

Cf. A000040, A003327.

Programs

Mathematica

mx = 1000; lim = Floor[(mx - 3)^(1/3)]; Select[Union[Total /@ Tuples[Range[lim]^3, {4}]], # <= mx && PrimeQ[#] &] (* Harvey P. Dale, May 25 2011 *)

Formula

A000040 INTERSECTION A003327.

Extensions

More terms from Harvey P. Dale, May 25 2011