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A307133 T(n,m) = number of k <= A002110(n) such that A001221(k) = m, where k is a term in A025487.

%I A307133 #9 Mar 27 2019 10:10:34
%S A307133 1,1,1,1,2,1,1,4,3,1,1,7,9,4,1,1,11,21,15,5,1,1,14,38,36,18,6,1,1,18,
%T A307133 64,79,53,23,7,1,1,23,97,148,122,63,26,7,1,1,27,140,258,251,157,76,30,
%U A307133 7,1,1,32,196,425,480,349,195,89,33,8,1,1,37,261,655,853
%N A307133 T(n,m) = number of k <= A002110(n) such that A001221(k) = m, where k is a term in A025487.
%C A307133 Terms m in A025487 are products of p_i# in A002110.
%C A307133 The primorial A002110(n) is the smallest number k that is the product of the n smallest primes (i.e., A001221(k) = n) and is a subset of A025487.
%H A307133 Michael De Vlieger, <a href="/A307133/b307133.txt">Table of n, a(n) for n = 0..560</a> (rows 0 <= n <= 1000).
%F A307133 T(n,0) = T(n,n) = A000012(n).
%F A307133 T(n,1) = A054850(n).
%F A307133 A098719(n) = sum of row n.
%e A307133 Row 3 = {1,4,3,1}. The terms k in A025487 such that k <= A002110(3) are {1, 2, 4, 6, 8, 12, 16, 24, 30}. Of these, 1 has 0 distinct prime divisors, 4 {2,4,8,16} have 1 distinct prime divisor, 3 {6,12,24} have 2 distinct prime divisors, and 1 {30} has 3 distinct prime divisors.
%e A307133 Triangle begins:
%e A307133    0: 1
%e A307133    1: 1   1
%e A307133    2: 1   2    1
%e A307133    3: 1   4    3    1
%e A307133    4: 1   7    9    4     1
%e A307133    5: 1  11   21   15     5     1
%e A307133    6: 1  14   38   36    18     6    1
%e A307133    7: 1  18   64   79    53    23    7    1
%e A307133    8: 1  23   97  148   122    63   26    7    1
%e A307133    9: 1  27  140  258   251   157   76   30    7    1
%e A307133   10: 1  32  196  425   480   349  195   89   33    8   1
%e A307133   11: 1  37  261  655   853   700  443  228  102   37   9   1
%e A307133   12: 1  42  340  975  1438  1323  928  533  268  119  41  11   1
%e A307133   ...
%t A307133 Block[{nn = 12, f, w}, f[n_] := {{1}}~Join~Block[{lim = Product[Prime@ i, {i, n}], ww = NestList[Append[#, 1] &, {1}, n - 1], g}, g[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]]; Map[Block[{w = #, k = 1}, Sort@ Prepend[If[Length@ # == 0, #, #[[1]]], Product[Prime@ i, {i, Length@ w}]] &@ Reap[Do[If[# < lim, Sow[#]; k = 1, If[k >= Length@ w, Break[], k++]] &@ g@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@ #] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1]]], {i, Infinity}]][[-1]]] &, ww]]; s = MapAt[Flatten, f@ nn, 1]; Array[Function[P, TakeWhile[Map[Count[#, _?(# <= P &)] &, s, {1}], # > 0 &]]@ Product[Prime@ i, {i, #}] &, nn + 1, 0]] // Flatten
%Y A307133 Cf. A000012, A001221, A002110, A025487, A054850, A098719.
%K A307133 nonn,tabl
%O A307133 0,5
%A A307133 _Michael De Vlieger_, Mar 26 2019