A210528 - OEIS

%I #15 Dec 05 2018 06:13:26

%S 1,1,1,2,2,2,2,2,2,3,3,4,3,3,3,5,3,6,2,5,4,6,3,7,3,5,4,5,3,8,3,6,6,7,

%T 4,9,4,6,6,6,3,10,4,7,8,8,3,12,4,7,9,8,4,12,5,10,8,9,4,14,3,9,8,11,4,

%U 13,4,11,9,9,4,15,4,10,9,11,5,13,4,12,11,11,5,17,4,10,11,11,4,15,4,12,11,11,3,16,3,11,12,13

%N Number of ways to write 2n = p+q (p<=q) with p, q and p^6+q^6 all practical.

%C Conjecture: a(n)>0 for all n>0. Moreover, for any positive integers m and n, if m is greater than one or n is not among 17, 181, 211, 251, 313, 337, then 2n can be written as p+q with p, q and p^{3m}+q^{3m} all practical.

%C This conjecture implies that for each m=1,2,3,... there are infinitely many practical numbers of the form p^{3m}+q^{3m} with p and q both practical.

%H Zhi-Wei Sun, <a href="/A210528/b210528.txt">Table of n, a(n) for n = 1..10000</a>

%H G. Melfi, <a href="http://dx.doi.org/10.1006/jnth.1996.0012">On two conjectures about practical numbers</a>, J. Number Theory 56 (1996) 205-210 [<a href="http://www.ams.org/mathscinet-getitem?mr=1370203">MR96i:11106</a>].

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2017.

%e a(3)=1 since 2*3=2+4 with 2, 4 and 2^6+4^6=4160 all practical.

%t f[n_]:=f[n]=FactorInteger[n]

%t Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])

%t Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]

%t pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)

%t a[n_]:=a[n]=Sum[If[pr[k]==True&&pr[2n-k]==True&&pr[k^6+(2n-k)^6]==True,1,0],{k,1,n}]

%t Do[Print[n," ",a[n]],{n,1,100}]

%Y Cf. A005153, A002372, A208244, A208246, A209253, A209254, A209315, A209320.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Jan 27 2013