OFFSET
1,1
COMMENTS
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.
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.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..77
Zhi-Wei Sun, Conjectures on representations involving primes, 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.
EXAMPLE
a(1) = 2 since 2 is prime and 2^2 - 2 = 2 is practical.
a(2) = 3 since 3 is prime and 2^3 - 2 = 6 is practical.
MATHEMATICA
f[n_]:=f[n]=FactorInteger[n];
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);
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}];
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);
tab={}; Do[If[pr[2^(Prime[k])-2], tab=Append[tab, Prime[k]]], {k, 1, 71}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 24 2017
STATUS
approved