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A122942 Partial product of n-th n-almost prime (A101695) divided by product of the first n primes, rounded down.

%I A122942 #22 Sep 02 2024 13:04:40
%S A122942 1,2,7,41,403,6960,196527,13405218,1566662070,304256578608,
%T A122942 113065670502087,78229220671714544,101598769325059903837,
%U A122942 293965406612712860369329,1613982664799943153033715558
%N A122942 Partial product of n-th n-almost prime (A101695) divided by product of the first n primes, rounded down.
%F A122942 a(n) = floor( Product_{i=1..n} A101695(i) / A000040(i) ).
%e A122942 a(1) = floor(2/2) = floor(1) = 1.
%e A122942 a(2) = floor(12/6) = floor(2) = 2.
%e A122942 a(3) = floor(216/30) = floor(7.2) = 7.
%e A122942 a(4) = floor(8640/210) = floor(41.1428571) = 41.
%e A122942 a(5) = floor(933120/2310) = floor(403.948052) = 403.
%e A122942 a(6) = floor(209018880/30030) = floor(6960.33566) = 6960.
%t A122942 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 A122942 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]; AlmostPrime[1, 1] = 2;
%t A122942 t = Table[ AlmostPrime[n, n], {n, 20}]; Floor[Rest@ FoldList[Times, 1, t]/Rest@ FoldList[Times, 1, Prime@ Range@ 20]] (* _Robert G. Wilson v_, Aug 31 2007 *)
%Y A122942 Cf. A101695, A000040.
%K A122942 easy,nonn,less
%O A122942 1,2
%A A122942 _Jonathan Vos Post_, Oct 24 2006
%E A122942 More terms from _Robert G. Wilson v_, Aug 31 2007