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A256383 Numbers n such that n-5 and n+5 are semiprimes.

%I A256383 #17 Sep 08 2022 08:46:11
%S A256383 9,20,30,44,60,82,90,116,124,128,138,150,164,182,208,210,214,242,254,
%T A256383 294,296,300,304,314,324,334,360,366,376,386,398,408,412,422,432,442,
%U A256383 476,506,510,522,524,532,538,540,548,578,584,586,628,674,676,684
%N A256383 Numbers n such that n-5 and n+5 are semiprimes.
%C A256383 It appears that there are no primes in this sequence.
%C A256383 If n is odd, one of n+5 and n-5 is divisible by 4, so unless n = 9 it can't be a semiprime.  Thus all terms except 9 are even. - _Robert Israel_, Apr 13 2020
%H A256383 Robert Israel, <a href="/A256383/b256383.txt">Table of n, a(n) for n = 1..10000</a>
%p A256383 N:= 1000: # for terms <= N-5
%p A256383 PP:= select(isprime, {seq(i,i=3..N/3,2)}):
%p A256383 P:= select(`<=`,PP,floor(sqrt(N))):
%p A256383 SP:= {}:
%p A256383 for p in P do
%p A256383   PP:= select(`<=`,PP,N/p);
%p A256383   SP:= SP union map(`*`,PP,p);
%p A256383 od:
%p A256383 R:= {9} union (map(`+`,SP,5) intersect map(`-`,SP,5)):
%p A256383 sort(convert(R,list)); # _Robert Israel_, Apr 13 2020
%t A256383 Select[Range[2, 700], PrimeOmega[# + 5] == PrimeOmega[# - 5] == 2 &] (* _Vincenzo Librandi_, Mar 29 2015 *)
%o A256383 (PARI) lista(nn,m=5) = {for (n=m+1, nn, if (bigomega(n-m)==2 && bigomega(n+m)==2, print1(n, ", ")););}
%o A256383 (PARI) issemi(n)=bigomega(n)==2
%o A256383 list(lim)=my(v=List([9])); forprime(p=5,(lim-5)\3, if(issemi(3*p+10), listput(v,3*p+5))); forprime(p=29,(lim+5)\3, if(issemi(3*p-10), listput(v,3*p-5))); forstep(n=30,lim\=1,6, if(issemi(n-5) && issemi(n+5), listput(v, n))); Set(v) \\ _Charles R Greathouse IV_, Apr 13 2020
%o A256383 (Magma) IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [6..700] | IsSemiprime(n+5) and IsSemiprime(n-5) ]; // _Vincenzo Librandi_, Mar 29 2015
%Y A256383 Cf. A001358 (semiprimes).
%Y A256383 Cf. A124936 (n-1 and n+1), A105571 (n-2 and n+2).
%Y A256383 Cf. A256381 (n-3 and n+3), A256382 (n-4 and n+4).
%K A256383 nonn
%O A256383 1,1
%A A256383 _Michel Marcus_, Mar 27 2015