cp's OEIS Frontend

A294200 Primes p such that 2^p - 2 is a practical number.

%I A294200 #12 Oct 25 2017 09:18:07
%S A294200 2,3,5,7,11,13,17,19,29,31,37,41,43,53,61,67,71,73,79,89,97,101,103,
%T A294200 109,113,127,131,137,139,151,157,163,181,191,193,197,199,211,223,229,
%U A294200 233,241,251,257,271,277,281,283,307,311,313,331,337,349,353
%N A294200 Primes p such that 2^p - 2 is a practical number.
%C A294200 Conjecture: The sequence has infinitely many terms. In other words, there are infinitely many practical numbers of the form 2^p - 2 with p prime.
%C A294200 By Fermat's little theorem, p divides 2^p - 2 for any prime p. Note that those 2^p - 1 with p prime are called Mersenne numbers.
%H A294200 Zhi-Wei Sun, <a href="/A294200/b294200.txt">Table of n, a(n) for n = 1..77</a>
%H A294200 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 A294200 a(1) = 2 since 2 is prime and 2^2 - 2 = 2 is practical.
%e A294200 a(2) = 3 since 3 is prime and 2^3 - 2 = 6 is practical.
%t A294200 f[n_]:=f[n]=FactorInteger[n];
%t A294200 Pow[n_, i_]:=Pow[n,i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);
%t A294200 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 A294200 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);
%t A294200 tab={};Do[If[pr[2^(Prime[k])-2],tab=Append[tab,Prime[k]]],{k,1,71}];Print[tab]
%Y A294200 Cf. A000040, A000043, A000079, A005153, A294112.
%K A294200 nonn
%O A294200 1,1
%A A294200 _Zhi-Wei Sun_, Oct 24 2017