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%I #45 Sep 08 2022 08:46:16
%S 10,14,15,21,26,33,35,38,39,51,62,65,69,77,86,91,93,95,111,122,123,
%T 129,133,146,159,161,201,203,206,209,213,215,217,218,221,249,278,287,
%U 291,299,301,302,303,305,321,335,339,362,371,381,386,395,398,403
%N Squarefree semiprimes n such that phi(n) - 1 is prime.
%C Equals (A001358 intersection A078892) - A001248.
%H Charles R Greathouse IV, <a href="/A271568/b271568.txt">Table of n, a(n) for n = 1..10000</a>
%e 15 is in the sequence, because 15 = 3*5 is a semiprime with omega(15) = 2 and phi(15) - 1 = 2*4 - 1 = 7 is a prime.
%e 21 is in the sequence, because 21 = 3*7 is a semiprime with omega(21) = 2 and phi(21) - 1 = 2*6 - 1 = 11 is a prime.
%p with(numtheory):
%p is_A271568 := n -> issqrfree(n) and bigomega(n) = 2 and isprime(phi(n)-1):
%p select(is_A271568, [$1..403]); # _Peter Luschny_, Jul 21 2016
%t A271568Q = SquareFreeQ[#] && PrimeNu[#] == 2 && PrimeQ[EulerPhi[#] - 1] &; Select[Range[500], A271568Q] (* _JungHwan Min_, Jul 29 2016 *)
%o (PARI) is_a001358(n) = bigomega(n)==2
%o is_a005117(n) = issquarefree(n)
%o is_a078892(n) = ispseudoprime(eulerphi(n)-1)
%o is(n) = is_a001358(n) && is_a005117(n) && is_a078892(n) \\ _Felix Fröhlich_, Jul 21 2016
%o (PARI) is(n)=my(f=factor(n)); f[,2]==[1,1]~ && isprime((f[1,1]-1)*(f[2,1]-1)-1) \\ _Charles R Greathouse IV_, Jul 21 2016
%o (PARI) list(lim)=my(v=List()); forprime(p=2, sqrt(lim), forprime(q=p+1, lim\p, if(isprime((p-1)*(q-1)-1), listput(v, p*q)))); Set(v) \\ _Charles R Greathouse IV_, Aug 29 2016
%o (Magma) [n: n in [1..500] |(EulerPhi(n)+DivisorSigma(1,n)) eq 2*(n+1) and IsPrime(EulerPhi(n)-1)]; // _Vincenzo Librandi_, Jul 29 2016
%Y Cf. A001358, A078892, A001248.
%K nonn
%O 1,1
%A _Debapriyay Mukhopadhyay_, Jul 13 2016
%E New name from _Charles R Greathouse IV_, Jul 29 2016
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