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A133796 a(n) = n-th prime + n-th semiprime.

%I A133796 #21 May 20 2023 06:43:01
%S A133796 6,9,14,17,25,28,38,41,48,55,64,71,76,81,86,99,108,112,122,128,131,
%T A133796 141,148,158,171,178,185,192,195,200,218,224,231,234,255,262,272,281,
%U A133796 286,294,301,304,320,326,331,340,353,366,372,375,388,397,400,412,423,432
%N A133796 a(n) = n-th prime + n-th semiprime.
%H A133796 Alois P. Heinz, <a href="/A133796/b133796.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Harvey P. Dale)
%F A133796 a(n) = A000040(n) + A001358(n).
%e A133796 a(1) = 1st prime + 1st semiprime = 2 + 4 = 6.
%e A133796 a(2) = 2nd prime + 2nd semiprime = 3 + 6 = 9.
%e A133796 a(3) = 3rd prime + 3rd semiprime = 5 + 9 = 14.
%p A133796 A000040 := proc(n) ithprime(n) ; end: A001358 := proc(n) option remember ; local a ; if n = 1 then 4 ; else for a from A001358(n-1)+1 do if numtheory[bigomega](a) = 2 then RETURN(a) ; fi ; od: fi ; end: A133796 := proc(n) A000040(n)+A001358(n) ; end: seq(A133796(n),n=1..100) ; # _R. J. Mathar_, Jan 07 2008
%t A133796 SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]; SemiPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[SemiPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; f[n_] := Prime@n + SemiPrime@n; Array[f, 56] (* _Robert G. Wilson v_ *)
%t A133796 Module[{nn=300,pr,semi,len},pr=Prime[Range[PrimePi[nn]]];semi=Select[ Range[ nn], PrimeOmega[#]==2&];len=Min[Length[pr],Length[semi]];Total/@ Thread[{Take[pr,len],Take[semi,len]}]] (* _Harvey P. Dale_, Jun 27 2014 *)
%o A133796 (PARI) A133796(n)=prime(n)+A001358(n) \\ _M. F. Hasler_, Apr 24 2019
%Y A133796 Cf. A000040, A001358.
%K A133796 easy,nonn
%O A133796 1,1
%A A133796 _Jonathan Vos Post_, Jan 05 2008
%E A133796 Corrected and extended by _Robert G. Wilson v_ and _R. J. Mathar_, Jan 05 2008
%E A133796 Typo in first Mma program corrected by _Giovanni Resta_, Jun 12 2016