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A346917
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Numbers that are a sum of the cubes of four primes, not necessarily distinct.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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seq[max_] := Module[{s = Select[Range[Floor @ Surd[max, 3]], PrimeQ]}, Select[Union[Plus @@@ (Tuples[s, 4]^3)], # <= max &]]; seq[2000]
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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