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A319386 Semiprimes k = pq with primes p < q such that p-1 does not divide q-1. 2

%I

%S 35,55,77,95,115,119,143,155,161,187,203,209,215,221,235,247,253,287,

%T 295,299,319,323,329,335,355,371,377,391,395,403,407,413,415,437,473,

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

%C 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 All the terms are odd and indivisible by 3.

%C If k is in the sequence, then gcd(k,b^k-b)=1 for some integer b.

%C These numbers are probably all semiprimes in A121707.

%H Robert Israel, <a href="/A319386/b319386.txt">Table of n, a(n) for n = 1..10000</a>

%e 35 = 5*7 is a term since 5-1 does not divide 7-1.

%e 35 is a term since lpf(35)-1 = 5-1 does not divide 35-1.

%p N:= 1000: # for terms <= N

%p P:= select(isprime,{seq(i,i=5..N/5,2)}):

%p S:= {}:

%p for p in P do

%p Qs:= select(q -> q > p and q <= N/p and (q-1 mod (p-1) <> 0), P);

%p S:= S union map(`*`,Qs,p);

%p od:

%p sort(convert(S,list)); # _Robert Israel_, Apr 14 2020

%o (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 (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 Subsequence of A046388.

%Y Complement of A162730 w.r.t. A006881.

%Y Cf. A001358, A121707.

%K nonn

%O 1,1

%A _Thomas Ordowski_, Sep 18 2018

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Last modified May 8 19:28 EDT 2021. Contains 343666 sequences. (Running on oeis4.)