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

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, 64, 79, 53, 23, 7, 1, 1, 23, 97, 148, 122, 63, 26, 7, 1, 1, 27, 140, 258, 251, 157, 76, 30, 7, 1, 1, 32, 196, 425, 480, 349, 195, 89, 33, 8, 1, 1, 37, 261, 655, 853

Offset: 0

Author

Michael De Vlieger, Mar 26 2019

Keywords

• nonn
• tabl

Comments

Terms m in A025487 are products of p_i# in A002110.

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.

Examples

			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.
Triangle begins:
   0: 1
   1: 1   1
   2: 1   2    1
   3: 1   4    3    1
   4: 1   7    9    4     1
   5: 1  11   21   15     5     1
   6: 1  14   38   36    18     6    1
   7: 1  18   64   79    53    23    7    1
   8: 1  23   97  148   122    63   26    7    1
   9: 1  27  140  258   251   157   76   30    7    1
  10: 1  32  196  425   480   349  195   89   33    8   1
  11: 1  37  261  655   853   700  443  228  102   37   9   1
  12: 1  42  340  975  1438  1323  928  533  268  119  41  11   1
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..560 (rows 0 <= n <= 1000).

Crossrefs

Cf. A000012, A001221, A002110, A025487, A054850, A098719.

Programs

Mathematica

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

Formula

T(n,0) = T(n,n) = A000012(n).

T(n,1) = A054850(n).

A098719(n) = sum of row n.