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aprQ[p_]:=Total[FactorInteger[#][[All, 2]]]&/@Range[p+1, p+4]=={2, 3, 4, 5}; Select[ Prime[ Range[120000]], aprQ] (* Harvey P. Dale, Dec 17 2022 *)
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(MAGMAMagma) [n: n in PrimesUpTo(2*10^6) | forall{k: k in [1..4] | &+[f[j, 2]: j in [1..#f]] eq k+1 where f is Factorization(n+k)}]; // Vincenzo Librandi, Sep 24 2012
Charles R Greathouse IV, <a href="/A113008/b113008_1.txt">Table of n, a(n) for n = 1..10000</a>
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Vincenzo Librandi, Charles R Greathouse IV, <a href="/A113008/b113008_1.txt">Table of n, a(n) for n = 1..100010000</a>
(PARI) list(lim)=my(v=List(), L=(lim+2)\3, t); forprime(p=3, L\3, forprime(q=3, min(L\p, p), t=3*p*q-2; if(t%12==1 && isprime(t) && isprime((t+1)/2) && bigomega(t+3)==4 && bigomega(t+4)==5, listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 05 2017
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f[n_] := Plus @@ Last /@ FactorInteger@n; t = {}; Do[p = Prime[n]; If[Array[ f[p + # ] &, 4] == {2, 3, 4, 5}, AppendTo[t, p]], {n, 126483}]; t (* _Robert G. Wilson v _ *)
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(MAGMA) [n: n in PrimesUpTo(2*10^6) | forall{k: k in [1..4] | &+[f[j, 2]: j in [1..#f]] eq k +1 where f is Factorization(n +k)}]; // Vincenzo Librandi, Sep 24 2012
proposed
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proposed