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A114791 Consider the array T where the element at T(r,c) is the number of n-almost primes less than or equal to 2^m. Sequence read by successive columns beginning at r=c.

%I A114791 #8 Sep 02 2024 13:04:36
%S A114791 1,1,2,1,2,4,1,2,6,6,1,2,7,10,11,1,2,7,13,22,18,1,2,7,14,30,42,31,1,2,
%T A114791 7,15,34,60,82,54,1,2,7,15,36,71,125,157,97,1,2,7,15,37,77,152,256,
%U A114791 304,172,1,2,7,15,37,81,168,325,513,589,309,1,2,7,15,37,83,177,367,669,1049
%N A114791 Consider the array T where the element at T(r,c) is the number of n-almost primes less than or equal to 2^m. Sequence read by successive columns beginning at r=c.
%C A114791 Eventually each column reads 1,2,7,15,37,84,187,421,914,... = A052130.
%F A114791 Each column sums to 2^n-1.
%e A114791 1 2 4 6 11 18 31 54 97 172 309 564 1028 1900 3512 6542
%e A114791 ..1 2 6 10 22 42 82 157 304 589 1124 2186 4192 8110 15658
%e A114791 ....1 2 7 13 30 60 125 256 513 1049 2082 4214 8401 16771
%e A114791 ......1 2 7 14 34 71 152 325 669 1405 2866 5931 12139
%e A114791 ........1 2 7 15 36 77 168 367 770 1643 3410 7150
%e A114791 ..........1 2 7 15 37 81 177 392 831 1790 3757
%e A114791 ............1 2 7 15 37 83 182 406 867 1880
%e A114791 ..............1 2 7 15 37 84 185 414 887
%e A114791 ................1 2 7 15 37 84 186 418
%t A114791 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 A114791 Table[ AlmostPrimePi[n, 2^k], {k, 12}, {n, k, 1, -1}] // Flatten
%Y A114791 First row is A007053, Cf. A052130.
%K A114791 nonn,tabl
%O A114791 1,3
%A A114791 _Robert G. Wilson v_, Feb 21 2006