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A306737 Irregular triangle where row n is a list of indices in A002110 with multiplicity whose product is A002182(n). 2
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 (list; graph; refs; listen; history; text; internal format)
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 A110627

Adjacent sequences:  A306734 A306735 A306736 * A306738 A306739 A306740

KEYWORD

nonn,tabf

AUTHOR

Michael De Vlieger, Mar 06 2019

STATUS

approved

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Last modified September 26 16:06 EDT 2021. Contains 347668 sequences. (Running on oeis4.)