A306737 Irregular triangle where row n is a list of indices in A002110 with multiplicity whose product is A002182(n).

0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 3, 2, 3, 1, 1, 1, 3, 1, 2, 3, 1, 1, 2, 3, 1, 1, 4, 2, 4, 1, 1, 1, 4, 1, 2, 4, 1, 1, 2, 4, 2, 2, 4, 1, 1, 1, 2, 4, 1, 2, 2, 4, 1, 1, 1, 1, 2, 4, 1, 1, 3, 4, 1, 2, 5, 2, 2, 2, 4, 1, 1, 1, 3, 4, 1, 1, 2, 5, 2, 2, 5, 1, 1, 1, 2, 5, 1, 2, 2, 5, 1, 1, 1, 1, 2, 5

Offset: 1

Author

Michael De Vlieger, Mar 06 2019

Keywords

• nonn
• tabf

Comments

Each highly composite number A002182(n) can be expressed as a product of primorials in A002110.

Row 1 = {0} by convention.

Maximum value in row n is given by A001221(A002182(n)).

Row n in reverse order is the conjugate of A067255(A002182(n)), a list of the multiplicities of the prime divisors of A002182(n).

Examples

			Terms in the first rows n of this sequence, followed by the corresponding primorials whose product = A002182(n):
   n    T(n,k)       A002110(T(n,k))   A002182(n)
  -----------------------------------------------
   1:   0;              1                =     1
   2:   1;              2                =     2
   3:   1, 1;           2 * 2            =     4
   4:   2;              6                =     6
   5:   1, 2;           2 * 6            =    12
   6:   1, 1, 2;        2 * 2 * 6        =    24
   7:   2, 2;           6 * 6            =    36
   8:   1, 1, 1, 2;     2 * 2 * 2 * 6    =    48
   9:   1, 3;           2 * 30           =    60
  10:   1, 1, 3;        2 * 2 * 30       =   120
  11:   2, 3;           6 * 30           =   180
  12:   1, 1, 1, 3;     2 * 2 * 2 * 30   =   240
  13:   1, 2, 3;        2 * 6 * 30       =   360
  14:   1, 1, 2, 3;     2 * 2 * 6 * 30   =   720
  15:   1, 1, 4;        2 * 2 * 210      =   840
  ...
Row 6 = {1,1,2} since A002110(1)*A002110(1)*A002110(2) = 2*2*6 = 24 and A002182(6) = 24. The conjugate of {2,1,1} = {3,1} and 24 = 2^3 * 3^1.
Row 10 = {1,1,3} since A002110(1)*A002110(1)*A002110(3) = 2*2*30 = 120 and A002182(10) = 120. The conjugate of {3,1,1} = {3,1,1} and 120 = 2^3 * 3^1 * 5^1.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10198, rows 1 <= n <= 1200, flattened.
• Michael De Vlieger, Charts showing terms in A002182 as a product of terms in A002110.
• Michael De Vlieger, Condensed text table showing terms in rows 1 <= n <= 10000.
• Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola (2018) Vol. 54, Issue 3.

Crossrefs

Cf. A001221, A002110, A002182, A067255, A304886.

Programs

Mathematica

With[{s = DivisorSigma[0, Range[250000]]}, Map[Reverse@ Table[LengthWhile[#, # >= i &], {i, Max@ #}] &@ If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #] &@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]] /. {} -> {0}] // Flatten