cp's OEIS Frontend

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

%I A346917 #11 Apr 09 2022 13:22:56
%S A346917 32,51,70,89,108,149,168,187,206,266,285,304,367,383,386,402,405,424,
%T A346917 484,500,503,522,601,620,702,718,721,740,819,838,936,1037,1056,1154,
%U A346917 1355,1372,1374,1393,1412,1472,1491,1510,1589,1608,1690,1706,1709,1728,1807,1826
%N A346917 Numbers that are a sum of the cubes of four primes, not necessarily distinct.
%C A346917 Roth (1951) proved that the number of terms below x is >> x/log(x)^8.
%C A346917 Ren (2001) proved that this sequence has a positive lower density.
%C A346917 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).
%H A346917 Michael S. Branicky, <a href="/A346917/b346917.txt">Table of n, a(n) for n = 1..10000</a>
%H A346917 Christian Elsholtz and Jan-Christoph Schlage-Puchta, <a href="https://arxiv.org/abs/1902.09858">The density of integers representable as the sum of four prime cubes</a>, arXiv preprint, arXiv:1902.09858 [math.NT], 2019.
%H A346917 Zhixin Liu, <a href="http://dx.doi.org/10.1016/j.jnt.2011.12.003">Density of the sums of four cubes of primes</a>, Journal of Number Theory, Vol. 132, No. 4 (2012), pp. 735-747.
%H A346917 Xiumin Ren, <a href="https://doi.org/10.1142/S025295990100022X">Density of integers that are the sum of four cubes of primes</a>, Chin. Ann. Math. Ser. B, Vol. 22, No. 2 (2001), pp. 233-242.
%H A346917 Xiumin Ren, <a href="https://doi.org/10.1016/S0022-314X(02)00022-7">Sums of four cubes of primes</a>, J. Number Theory, Vol. 98, No. 1 (2003), pp. 156-171.
%H A346917 K. F. Roth, <a href="https://doi.org/10.1112/plms/s2-53.4.268">On Waring's problem for cubes</a>, Proc. London Math. Soc. (2), Vol. 53 (1951), pp. 268-279.
%e A346917 a(1) = 32 = 2^3 + 2^3 + 2^3 + 2^3.
%e A346917 a(2) = 51 = 2^3 + 2^3 + 2^3 + 3^3.
%e A346917 a(3) = 70 = 2^3 + 2^3 + 3^3 + 3^3.
%t A346917 seq[max_] := Module[{s = Select[Range[Floor @ Surd[max, 3]], PrimeQ]}, Select[Union[Plus @@@ (Tuples[s, 4]^3)], # <= max &]]; seq[2000]
%o A346917 (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
%o A346917 (Python)
%o A346917 from sympy import integer_nthroot, primerange
%o A346917 from itertools import combinations_with_replacement as cwr
%o A346917 def aupto(limit):
%o A346917     cubes = [p**3 for p in primerange(2, integer_nthroot(limit, 3)[0])]
%o A346917     return sorted(sum(c) for c in cwr(cubes, 4) if sum(c) <= limit)
%o A346917 print(aupto(2000)) # _Michael S. Branicky_, Apr 09 2022
%Y A346917 Cf. A030078, A086119, A258865.
%K A346917 nonn
%O A346917 1,1
%A A346917 _Amiram Eldar_, Aug 07 2021