A346917 Numbers that are a sum of the cubes of four primes, not necessarily distinct.

32, 51, 70, 89, 108, 149, 168, 187, 206, 266, 285, 304, 367, 383, 386, 402, 405, 424, 484, 500, 503, 522, 601, 620, 702, 718, 721, 740, 819, 838, 936, 1037, 1056, 1154, 1355, 1372, 1374, 1393, 1412, 1472, 1491, 1510, 1589, 1608, 1690, 1706, 1709, 1728, 1807, 1826

Offset

1,1

Comments

Roth (1951) proved that the number of terms below x is >> x/log(x)^8.

Ren (2001) proved that this sequence has a positive lower density.

The lower density was proven to be larger than 0.003125 (Ren, 2003), 0.005776 (Liu, 2012), and 0.009664 (Elsholtz and Schlage-Puchta, 2019).

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Christian Elsholtz and Jan-Christoph Schlage-Puchta, The density of integers representable as the sum of four prime cubes, arXiv preprint, arXiv:1902.09858 [math.NT], 2019.

Zhixin Liu, Density of the sums of four cubes of primes, Journal of Number Theory, Vol. 132, No. 4 (2012), pp. 735-747.

Xiumin Ren, Density of integers that are the sum of four cubes of primes, Chin. Ann. Math. Ser. B, Vol. 22, No. 2 (2001), pp. 233-242.

Xiumin Ren, Sums of four cubes of primes, J. Number Theory, Vol. 98, No. 1 (2003), pp. 156-171.

K. F. Roth, On Waring's problem for cubes, Proc. London Math. Soc. (2), Vol. 53 (1951), pp. 268-279.

Example

a(1) = 32 = 2^3 + 2^3 + 2^3 + 2^3.
a(2) = 51 = 2^3 + 2^3 + 2^3 + 3^3.
a(3) = 70 = 2^3 + 2^3 + 3^3 + 3^3.

Mathematica

seq[max_] := Module[{s = Select[Range[Floor @ Surd[max, 3]], PrimeQ]}, Select[Union[Plus @@@ (Tuples[s, 4]^3)], # <= max &]]; seq[2000]

Prog

(PARI) list(lim)=my(v=List(), P=apply(p->p^3, primes(sqrtnint(lim\=1, 3)))); foreach(P, p, foreach(P, q, foreach(P, r, my(s=p+q+r, t); for(i=1, #P, t=s+P[i]; if(t>lim, break); listput(v, t))))); Set(v) \\ Charles R Greathouse IV, Aug 09 2021
(Python)
from sympy import integer_nthroot, primerange
from itertools import combinations_with_replacement as cwr
def aupto(limit):
    cubes = [p**3 for p in primerange(2, integer_nthroot(limit, 3)[0])]
    return sorted(sum(c) for c in cwr(cubes, 4) if sum(c) <= limit)
print(aupto(2000)) # Michael S. Branicky, Apr 09 2022

Crossrefs

Cf. A030078, A086119, A258865.

Sequence in context: A316943 A099048 A176542 * A343154 A048734 A277870

Adjacent sequences: A346914 A346915 A346916 * A346918 A346919 A346920

Keyword

nonn

Author

Amiram Eldar, Aug 07 2021

Status

approved