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%I #25 Apr 10 2019 12:46:02
%S 41,71,101,137,149,179,191,197,227,239,269,281,311,347,419,431,461,
%T 569,599,617,641,659,809,821,827,857,881,1019,1031,1049,1061,1091,
%U 1151,1229,1289,1301,1319,1427,1451,1481,1487,1607,1619,1667,1697,1721,1787,1871,1877
%N Lesser member p of twin primes in A054723 (Prime exponents of composite Mersenne numbers).
%C 2^p-1 is composite. p is the lesser of twin primes in A001359 and a prime exponent of a Mersenne number in A054723.
%H Harvey P. Dale, <a href="/A289982/b289982.txt">Table of n, a(n) for n = 1..1000</a>
%e p=41 is a member because 41 is a lesser of twin prime and 2^41 - 1 = 13367*164511353 is composite.
%e Similarly, p=227 is a member because 227 is a lesser of twin prime and 2^227 - 1 is composite.
%t 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 *)
%t 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 *)
%o (GAP)
%o 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]);;
%o P2:=List([1..Length(P1)-1],i->[P1[i],P1[i+1]]);;
%o P3:=List(Positions(List(P2,i->i[2]-i[1]),2),i->P2[i][1]);
%o (PARI) isok(n) = isprime(n) && isprime(n+2) && !isprime(2^n-1) && !isprime(2^(n+2)-1); \\ _Michel Marcus_, Jul 19 2017
%Y Subsequence of A054723.
%Y Cf. A001097, A001359, A006512, A077800.
%K nonn
%O 1,1
%A _Muniru A Asiru_, Jul 17 2017
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