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Primes p such that p is equivalent to 3 modulo 4, p is neither 11 nor 23, and p is not a generalized repunit prime (i.e., p cannot be written as (q^t-1)/(q-1) for any prime-power q).
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%I #20 Jul 16 2017 14:39:55

%S 19,43,47,59,67,71,79,83,103,107,131,139,151,163,167,179,191,199,211,

%T 223,227,239,251,263,271,283,311,331,347,359,367,379,383,419,431,439,

%U 443,463,467,479,487,491,499

%N Primes p such that p is equivalent to 3 modulo 4, p is neither 11 nor 23, and p is not a generalized repunit prime (i.e., p cannot be written as (q^t-1)/(q-1) for any prime-power q).

%C The numbers in this sequence are called zeta-primes, and they exactly identify when (for n > 4) the set of maximal subgroups of even order fail to cover Alt(n). This is proved in the reference below.

%H B. J. Benesh, D. C. Ernst, and N. Sieben <a href="http://arxiv.org/abs/1508.03419">Impartial avoidance and achievement games for generating symmetric and alternating groups</a>, arXiv:1508.03419 [math.CO], 2015.

%H H. Dubner, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1185243-9">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930.

%o (GAP)

%o # Primes is a list of the 168 primes below 1000.

%o primeList:=[];

%o primeList:=ShallowCopy(Primes);

%o # Remove {3} and {11,23}, which are in the 2nd, 5th, and 9th positions, respectively.

%o Remove(primeList,9);

%o Remove(primeList,5);

%o Remove(primeList,2);

%o # Remove anything that is not 3 mod 4.

%o primeList:=Filtered(primeList,p->p mod 4 = 3);

%o # This generates all repunits so that we may remove them from the list of primes.

%o repunitList:=[];

%o for q in [2..1000] do

%o if IsPrimePowerInt(q) then

%o n:=1;

%o x:=(q^n-1)/(q-1);

%o while x < 1000 do

%o Add(repunitList,x);

%o n:=n+1;

%o x:=(q^n-1)/(q-1);

%o od;

%o fi;

%o od;

%o # Remove repunits from filtered prime list to produce list of zeta-primes

%o getZeta:=function()

%o local zlist,p;

%o zlist:=[];

%o for p in primeList do

%o if not p in repunitList then

%o Add(zlist,p);

%o fi;

%o od;

%o return zlist;

%o end;

%Y Subsequence of A002145, A028491 gives examples of generalized repunit primes.

%K nonn

%O 1,1

%A _Bret Benesh_, Mar 11 2016