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A270123
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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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0
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19, 43, 47, 59, 67, 71, 79, 83, 103, 107, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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PROG
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(GAP)
# Primes is a list of the 168 primes below 1000.
primeList:=[];
primeList:=ShallowCopy(Primes);
# Remove {3} and {11, 23}, which are in the 2nd, 5th, and 9th positions, respectively.
Remove(primeList, 9);
Remove(primeList, 5);
Remove(primeList, 2);
# Remove anything that is not 3 mod 4.
primeList:=Filtered(primeList, p->p mod 4 = 3);
# This generates all repunits so that we may remove them from the list of primes.
repunitList:=[];
for q in [2..1000] do
if IsPrimePowerInt(q) then
n:=1;
x:=(q^n-1)/(q-1);
while x < 1000 do
Add(repunitList, x);
n:=n+1;
x:=(q^n-1)/(q-1);
od;
fi;
od;
# Remove repunits from filtered prime list to produce list of zeta-primes
getZeta:=function()
local zlist, p;
zlist:=[];
for p in primeList do
if not p in repunitList then
Add(zlist, p);
fi;
od;
return zlist;
end;
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CROSSREFS
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Subsequence of A002145, A028491 gives examples of generalized repunit primes.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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