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%I #5 Oct 27 2017 21:50:33
%S 169,625,729,1089,1681,3969,4225,5929,6241,6561,6889,8647,9409,11449,
%T 14641,15625,16129,18769,20449,22201,24649,27561,28561,30625,32761,
%U 33331,33489,33661,34969,35209,35721,38071,38809,39601,41209,42025,43681,43969,44521,47089,47961,50625,51529,55225,58081
%N Positive integers m with m^k - 1 (k = 1,...,13) all practical.
%C Conjecture: For any positive integer n, there are infinitely many positive integers m with m^k - 1 (k = 1,...,n) all practical.
%C This is true for n = 2. In fact, by a result of Melfi, there are infinitely many positive integers m such that m - 1 and m + 1 are both practical and hence (m-1)*(m+1) = m^2 - 1 is also practical.
%H Zhi-Wei Sun, <a href="/A294307/b294307.txt">Table of n, a(n) for n = 1..200</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) = 169 since 169 is the first number m such that m - 1, m^2 - 1, ..., m^13 - 1 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-1]&&pr[n^2-1]&&pr[n^3-1]&&pr[n^4-1]&&pr[n^5-1]&&pr[n^6-1]&&pr[n^7-1]&&pr[n^8-1]&&pr[n^9-1]&&pr[n^(10)-1]&&pr[n^(11)-1]&&pr[n^(12)-1]&&pr[n^(13)-1]
%t tab={};Do[If[pq[k],tab=Append[tab,k]],{k,1,59000}];Print[tab]
%Y Cf. A005153, A294112, A294200, A294225.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Oct 27 2017
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