2,2

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 ofA082288(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 byAlois P. HeinzinA215366as 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

FromEmeric 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)

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.

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

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

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.)

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

(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

(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

Row lengths areA001222.Cf.A000040,A027746,A000720,A036036.

Cf.A056239(row sums).

Cf.A003963(row products).

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

Sequence in context:A131375A358169A327392*A187846A181087A029288

Adjacent sequences:A112795A112796A112797*A112799A112800A112801

nonn,tabf

Franklin T. Adams-Watters,Jan 22 2006

approved