A112798 Table where n-th row is factorization of n, with each prime p_i replaced by i.

1, 2, 1, 1, 3, 1, 2, 4, 1, 1, 1, 2, 2, 1, 3, 5, 1, 1, 2, 6, 1, 4, 2, 3, 1, 1, 1, 1, 7, 1, 2, 2, 8, 1, 1, 3, 2, 4, 1, 5, 9, 1, 1, 1, 2, 3, 3, 1, 6, 2, 2, 2, 1, 1, 4, 10, 1, 2, 3, 11, 1, 1, 1, 1, 1, 2, 5, 1, 7, 3, 4, 1, 1, 2, 2, 12, 1, 8, 2, 6, 1, 1, 1, 3, 13, 1, 2, 4, 14, 1, 1, 5, 2, 2, 3, 1, 9, 15, 1, 1, 1, 1

Offset: 2

Author

Franklin T. Adams-Watters, Jan 22 2006

Keywords

• core
• nonn
• tabf

Comments

This is an enumeration of all partitions.

Technically this is an enumeration of all multisets (finite weakly increasing sequences of positive integers) rather than integer partitions. - Gus Wiseman, Dec 12 2016

A000040(a(n)) is a prime factor of A082288(n). - Reinhard Zumkeller, Feb 03 2008

Row n is the partition with Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1..r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. For a given n, the 2nd Maple program yields row n; for example, we obtain at once B(2436) = [1,1,2,4,10]. - Emeric Deutsch, Jun 04 2015

From Emeric Deutsch, May 05 2015: (Start)

Number of entries in row n is bigomega(n) (i.e., the number of prime factors of n, multiplicities included).

Product of entries in row n = A003963(n).

Row n contains the Matula numbers of the rooted trees obtained from the rooted tree with Matula number n by deleting the edges emanating from the root. Example: row 8 is 1,1,1; indeed the rooted tree with Matula number 8 is \|/ and deleting the edges emanating from the root we obtain three one-vertex trees, having Matula numbers 1, 1, 1. Example: row 7 is 4; indeed, the rooted tree with Matula number 7 is Y and deleting the edges emanating from the root we obtain the rooted tree V, having Matula number 4.

The Matula (or Matula-Goebel) number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m >= 2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. (End)

Examples

			Row 20 is 1,1,3 because the prime factors of 20, namely 2,2,5 are the 1st, 1st, 3rd primes.
Table begins:
  1;
  2;
  1, 1;
  3;
  1, 2;
  4;
  1, 1, 1;
  ...
The sequence of all finite multisets of positive integers begins: (), (1), (2), (11), (3), (12), (4), (111), (22), (13), (5), (112), (6), (14), (23), (1111), (7), (122), (8), (113), (24), (15), (9), (1112), (33), (16), (222), (114). - _Gus Wiseman_, Dec 12 2016
		

Links

• Alois P. Heinz, Rows n = 2..3275, flattened
• Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
• Emeric Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
• F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
• I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
• I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
• D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Row lengths are A001222. Cf. A000040, A027746, A000720, A036036.

Cf. A056239 (row sums).

Cf. A003963 (row products).

Cf. A049084, A082288, A215366, A241918, A265146, A275024.

Programs

Haskell

a112798 n k = a112798_tabf !! (n-2) !! (n-1)
a112798_row n = a112798_tabf !! (n-2)
a112798_tabf = map (map a049084) $ tail a027746_tabf
-- Reinhard Zumkeller, Aug 04 2014

Maple

T:= n-> sort([seq(numtheory[pi](i[1])$i[2], i=ifactors(n)[2])])[]:
seq(T(n), n=2..50);  # Alois P. Heinz, Aug 09 2012
with(numtheory): B := proc (n) local nn, j, m: nn := op(2, ifactors(n)); for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: # Emeric Deutsch, Jun 04 2015. (This is equivalent to the first Maple program.)

Mathematica

PrimePi /@ Flatten[Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ Range@ 60 // Flatten // Rest (* Michael De Vlieger, May 09 2015 *)

PARI

row(n)=my(v=List(),f=factor(n)); for(i=1,#f~,for(j=1,f[i,2], listput(v,primepi(f[i,1])))); Vec(v) \\ Charles R Greathouse IV, Nov 09 2021

Formula

T(n,k) = A000720(A027746(n,k)); A027746(n,k) = A000040(T(n,k)).

Also T(n,k) = A049084(A027746(n,k)). - Reinhard Zumkeller, Aug 04 2014