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A088179 Primes p such that mu(p-1) = 1; that is, p-1 is squarefree and has an even number of prime factors, where mu is the Moebius function.

%I A088179 #43 Feb 16 2025 08:32:51
%S A088179 2,7,11,23,47,59,83,107,167,179,211,227,263,331,347,359,383,463,467,
%T A088179 479,503,547,563,571,587,691,719,839,859,863,887,911,967,983,1019,
%U A088179 1123,1187,1231,1283,1291,1303,1307,1319,1327,1367,1439,1483,1487,1523,1619,1723
%N A088179 Primes p such that mu(p-1) = 1; that is, p-1 is squarefree and has an even number of prime factors, where mu is the Moebius function.
%C A088179 It is an unsolved problem to determine if this sequence has a positive density in the primes. - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003
%C A088179 Except for the initial element 2, this sequence seems to be exactly those primes the sum of whose nonquadratic, nonprimitive-root residues is congruent to -1(mod p). - _Dimitri Papadopoulos_, Jan 10 2016
%H A088179 Robert Israel, <a href="/A088179/b088179.txt">Table of n, a(n) for n = 1..10000</a>
%H A088179 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusFunction.html">Moebius Function</a>
%p A088179 filter:= proc(p) isprime(p) and numtheory:-mobius(p-1) = 1 end proc:
%p A088179 select(filter, [2,seq(i,i=3..2000,2)]); # _Robert Israel_, Feb 03 2016
%t A088179 Select[Prime[Range[400]], MoebiusMu[ #-1]==1&]
%o A088179 (PARI) lista(nn) = forprime(p=2, nn, if (moebius(p-1) == 1, print1(p, ", "))); \\ _Michel Marcus_, Jan 10 2016
%o A088179 (PARI) list(lim)=my(v=List(),last); forsquarefree(k=1,lim\1, if(moebius(k)==1, last=k[1], if(k[2][,2]==[1]~ && k[1]-last==1, listput(v,k[1])))); Vec(v) \\ _Charles R Greathouse IV_, Jan 08 2018
%o A088179 (Magma) [n: n in [2..2000] | IsPrime(n) and MoebiusMu(n-1) eq 1]; // _Vincenzo Librandi_, Jan 10 2016
%Y A088179 Cf. A049092 (primes p with mu(p-1)=0), A078330 (primes p with mu(p-1)=-1), A089451 (mu(p-1) for prime p).
%Y A088179 Cf. A002496.
%K A088179 nonn
%O A088179 1,1
%A A088179 _N. J. A. Sloane_ and _T. D. Noe_, Nov 03 2003