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

0,5

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.

Links

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

Formula

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

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

A098719(n) = sum of row n.

Example

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
  ...

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

Crossrefs

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

Sequence in context: A091186 A138155 A214986 * A218664 A247286 A055587

Adjacent sequences: A307130 A307131 A307132 * A307134 A307135 A307136

Keyword

nonn,tabl

Author

Michael De Vlieger, Mar 26 2019

Status

approved