%I #9 Mar 27 2019 10:10:34
%S 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 64,79,53,23,7,1,1,23,97,148,122,63,26,7,1,1,27,140,258,251,157,76,30,
%U 7,1,1,32,196,425,480,349,195,89,33,8,1,1,37,261,655,853
%N T(n,m) = number of k <= A002110(n) such that A001221(k) = m, where k is a term in A025487.
%C Terms m in A025487 are products of p_i# in A002110.
%C 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 Michael De Vlieger, <a href="/A307133/b307133.txt">Table of n, a(n) for n = 0..560</a> (rows 0 <= n <= 1000).
%F T(n,0) = T(n,n) = A000012(n).
%F T(n,1) = A054850(n).
%F A098719(n) = sum of row n.
%e 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 Triangle begins:
%e 0: 1
%e 1: 1 1
%e 2: 1 2 1
%e 3: 1 4 3 1
%e 4: 1 7 9 4 1
%e 5: 1 11 21 15 5 1
%e 6: 1 14 38 36 18 6 1
%e 7: 1 18 64 79 53 23 7 1
%e 8: 1 23 97 148 122 63 26 7 1
%e 9: 1 27 140 258 251 157 76 30 7 1
%e 10: 1 32 196 425 480 349 195 89 33 8 1
%e 11: 1 37 261 655 853 700 443 228 102 37 9 1
%e 12: 1 42 340 975 1438 1323 928 533 268 119 41 11 1
%e ...
%t 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 Cf. A000012, A001221, A002110, A025487, A054850, A098719.
%K nonn,tabl
%O 0,5
%A _Michael De Vlieger_, Mar 26 2019