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

1,5

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).

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.

Example

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.

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

Crossrefs

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

Sequence in context: A065373 A047895 A307322 * A178474 A164822 A361208

Adjacent sequences: A306734 A306735 A306736 * A306738 A306739 A306740

Keyword

nonn,tabf

Author

Michael De Vlieger, Mar 06 2019

Status

approved