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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified September 20 16:50 EDT 2021. Contains 347586 sequences. (Running on oeis4.)