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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Aug 07 2021
STATUS
approved