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

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 151, 157, 163, 181, 191, 193, 197, 199, 211, 223, 229, 233, 241, 251, 257, 271, 277, 281, 283, 307, 311, 313, 331, 337, 349, 353

Offset: 1

Author

Zhi-Wei Sun, Oct 24 2017

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000040, A000043, A000079, A005153, A294112.

Programs

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]