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