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A319386 Semiprimes k = pq with primes p < q such that p-1 does not divide q-1.

%I A319386 #22 Jun 06 2021 11:06:18
%S A319386 35,55,77,95,115,119,143,155,161,187,203,209,215,221,235,247,253,287,
%T A319386 295,299,319,323,329,335,355,371,377,391,395,403,407,413,415,437,473,
%U A319386 493,497,515,517,527,533,535,551,559,581,583,589,611,623,629,635,649,655,667,689,695,697,707,713,731
%N A319386 Semiprimes k = pq with primes p < q such that p-1 does not divide q-1.
%C A319386 The "anti-Carmichael semiprimes" defined: semiprimes k such that lpf(k)-1 does not divide k-1; then also gpf(k)-1 does not divide k-1.
%C A319386 All the terms are odd and indivisible by 3.
%C A319386 If k is in the sequence, then gcd(k,b^k-b)=1 for some integer b.
%C A319386 These numbers are probably all semiprimes in A121707.
%H A319386 Robert Israel, <a href="/A319386/b319386.txt">Table of n, a(n) for n = 1..10000</a>
%e A319386 35 = 5*7 is a term since 5-1 does not divide 7-1.
%e A319386 35 is a term since lpf(35)-1 = 5-1 does not divide 35-1.
%p A319386 N:= 1000: # for terms <= N
%p A319386 P:= select(isprime,{seq(i,i=5..N/5,2)}):
%p A319386 S:= {}:
%p A319386 for p in P do
%p A319386   Qs:= select(q -> q > p and q <= N/p and (q-1 mod (p-1) <> 0), P);
%p A319386   S:= S union map(`*`,Qs,p);
%p A319386 od:
%p A319386 sort(convert(S,list)); # _Robert Israel_, Apr 14 2020
%t A319386 spndQ[n_]:=Module[{fi=FactorInteger[n][[All,1]]},PrimeOmega[n]==2 && Length[ fi]==2&&Mod[fi[[2]]-1,fi[[1]]-1]!=0]; Select[Range[800],spndQ] (* _Harvey P. Dale_, Jun 06 2021 *)
%o A319386 (PARI) isok(n) = {if ((bigomega(n) == 2) && (omega(n) == 2), my(p = factor(n)[1, 1], q = factor(n)[2, 1]); (q-1) % (p-1) != 0;);}  \\ _Michel Marcus_, Sep 18 2018
%o A319386 (PARI) list(lim)=my(v=List(),s=sqrtint(lim\=1)); forprime(q=7,lim\5, forprime(p=5,min(min(q-2,s),lim\q), if((q-1)%(p-1), listput(v,p*q)))); Set(v) \\ _Charles R Greathouse IV_, Apr 14 2020
%Y A319386 Subsequence of A046388.
%Y A319386 Complement of A162730 w.r.t. A006881.
%Y A319386 Cf. A001358, A121707.
%K A319386 nonn
%O A319386 1,1
%A A319386 _Thomas Ordowski_, Sep 18 2018