A289982 Lesser member p of twin primes in A054723 (Prime exponents of composite Mersenne numbers).

41, 71, 101, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607, 1619, 1667, 1697, 1721, 1787, 1871, 1877

Offset: 1

Author

Muniru A Asiru, Jul 17 2017

Keywords

• nonn

Comments

2^p-1 is composite. p is the lesser of twin primes in A001359 and a prime exponent of a Mersenne number in A054723.

Examples

			p=41 is a member because 41 is a lesser of twin prime and 2^41 - 1 = 13367*164511353 is composite.
Similarly, p=227 is a member because 227 is a lesser of twin prime and 2^227 - 1 is composite.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Subsequence of A054723.

Cf. A001097, A001359, A006512, A077800.

Programs

GAP

P1:=Difference(Filtered([1..100000],IsPrime),[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701,23209, 44497, 86243]);;
P2:=List([1..Length(P1)-1],i->[P1[i],P1[i+1]]);;
P3:=List(Positions(List(P2,i->i[2]-i[1]),2),i->P2[i][1]);

Mathematica

Function[s, Flatten@ Map[s[[#, 1]] &, Position[Most@ s, d_ /; Quiet@ Differences@ d == {2}, {1}]]]@ Partition[#, 2, 1] &@ Select[Prime@ Range@ 360, ! PrimeQ[2^# - 1] &] (* Michael De Vlieger, Jul 17 2017 *)
Select[Partition[Module[{nn=20,mp},mp=MersennePrimeExponent[Range[nn]];Complement[Prime[Range[PrimePi[Last[mp]]]],mp]],2,1],#[[2]]-#[[1]]==2 && AllTrue[#,PrimeQ]&][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 10 2019 *)

PARI

isok(n) = isprime(n) && isprime(n+2) && !isprime(2^n-1) && !isprime(2^(n+2)-1); \\ Michel Marcus, Jul 19 2017