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%I #12 Oct 25 2017 17:37:11
%S 2,4,520,2560,3100,4648,6448,6784,7252,11128,12400,15496,19264,26128,
%T 26752,26860,28768,31648,32368,36160,37408,41728,45400,48760,53248,
%U 53584,54832,57148,58828,63544,66820,68440,68500,73948,74176,80512,81508,84208,93184,94300,106780,112288,113968,118528,131068
%N Practical numbers q with q + 2 and q^2 + 2 both practical.
%C Conjecture: The sequence has infinitely many terms.
%C In 1996 G. Melfi proved that there are infinitely many positive integers q with q and q + 2 both practical.
%C As any practical number greater than 2 is a multiple of 4 or 6, when q > 2, q + 2 and q^2 + 2 are all practical, we must have q^2 + 2 == 0 (mod 6), hence q is not divisible by 3 and thus 4 | q and 6 | (q + 2), therefore q == 4 (mod 12).
%H Zhi-Wei Sun, <a href="/A294225/b294225.txt">Table of n, a(n) for n = 1..2500</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.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures on representations involving primes</a>, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017.
%e a(1) = 2 since 2, 2 + 2 = 4 and 2^2 + 2 = 6 are all practical.
%e a(2) = 4 since 4, 4 + 2 = 6 and 4^2 + 2 = 18 are 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 pq[n_]:=pq[n]=pr[n]&&pr[n+2]&&pr[n^2+2];
%t tab={};Do[If[pq[k],tab=Append[tab,k]],{k,1,132000}];Print[tab]
%Y Cf. A005153, A287681, A294112, A294200.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Oct 25 2017
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