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%I #10 Aug 01 2015 11:53:39
%S 14,21,32,37,52,56,58,83,92,100,114,123,139,147,154,169,183,188,200,
%T 220,229,240,250,263,281,292,301,309,319,325,348,362,378,382,408,416,
%U 436,446,456,465,473,478,495,508,517,528,543,561,579,587,610,627,631,648
%N Prime(n) + Semiprime(n) + (3-almostPrime)(n).
%C Primes in this sequence begin: a(4) = 37, a(8) = 83, a(13) = 139, a(21) = 229, a(24) = 263, a(25) = 281, a(50) = 587. a(n) is semiprime for n = 1, 2, 12, 16, 17, 27, 28, 29, 32, 34, 38, 41, 42, 45, 47, 49. a(n) is 3-almost prime for n = 5, 7, 9, 11, 14, 15, 18, 26, 30, 40, 44, 48. This is the 3rd of an infinite superset of sequences Prime(n) + Semiprime(n) + (3-almostPrime)(n) ... + (k-almostPrime)(n).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F a(n) = A000040(n) + A001358(n) + A014612(n).
%e a(1) = Prime(1) + Semiprime(1) + 3AlmostPrime(1) = 2 + 4 + 8 = 14.
%e a(10) = Prime(10) + Semiprime(10) + 3AlmostPrime(10) = 29 + 26 + 45 = 100.
%e a(20) = Prime(20) + Semiprime(20) + 3AlmostPrime(20) = 71 + 57 + 92 = 220.
%e a(30) = Prime(30) + Semiprime(30) + 3AlmostPrime(30) = 113 + 87 + 125 = 325.
%e a(40) = Prime(40) + Semiprime(40) + 3AlmostPrime(40) = 173 + 121 + 71 = 465.
%e a(50) = Prime(50) + Semiprime(50) + 3AlmostPrime(50) = 229 + 146 + 212 = 587.
%t AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[ Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]] ]]]; (* _Eric W. Weisstein_, Feb 07 2006 *)
%t AlmostPrime[k_, n_] := Block[{e = Floor[Log[2, n]+k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Table[ Sum[ AlmostPrime[k, n], {k, 3}], {n, 54}] (* _Robert G. Wilson v_, Feb 21 2006 *)
%Y Cf. A000040, A001358, A014612.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Feb 11 2006
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