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%I #72 Jan 24 2024 21:01:55
%S 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,
%T 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,
%U 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
%N Table where n-th row is factorization of n, with each prime p_i replaced by i.
%C This is an enumeration of all partitions.
%C Technically this is an enumeration of all multisets (finite weakly increasing sequences of positive integers) rather than integer partitions. - _Gus Wiseman_, Dec 12 2016
%C A000040(a(n)) is a prime factor of A082288(n). - _Reinhard Zumkeller_, Feb 03 2008
%C 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
%C From _Emeric Deutsch_, May 05 2015: (Start)
%C Number of entries in row n is bigomega(n) (i.e., the number of prime factors of n, multiplicities included).
%C Product of entries in row n = A003963(n).
%C 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.
%C 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)
%H Alois P. Heinz, <a href="/A112798/b112798.txt">Rows n = 2..3275, flattened</a>
%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H Emeric Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.
%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F T(n,k) = A000720(A027746(n,k)); A027746(n,k) = A000040(T(n,k)).
%F Also T(n,k) = A049084(A027746(n,k)). - _Reinhard Zumkeller_, Aug 04 2014
%e Row 20 is 1,1,3 because the prime factors of 20, namely 2,2,5 are the 1st, 1st, 3rd primes.
%e Table begins:
%e 1;
%e 2;
%e 1, 1;
%e 3;
%e 1, 2;
%e 4;
%e 1, 1, 1;
%e ...
%e 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
%p T:= n-> sort([seq(numtheory[pi](i[1])$i[2], i=ifactors(n)[2])])[]:
%p seq(T(n), n=2..50); # _Alois P. Heinz_, Aug 09 2012
%p 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.)
%t PrimePi /@ Flatten[Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ Range@ 60 // Flatten // Rest (* _Michael De Vlieger_, May 09 2015 *)
%o (Haskell)
%o a112798 n k = a112798_tabf !! (n-2) !! (n-1)
%o a112798_row n = a112798_tabf !! (n-2)
%o a112798_tabf = map (map a049084) $ tail a027746_tabf
%o -- _Reinhard Zumkeller_, Aug 04 2014
%o (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
%Y Row lengths are A001222. Cf. A000040, A027746, A000720, A036036.
%Y Cf. A056239 (row sums).
%Y Cf. A003963 (row products).
%Y Cf. A049084, A215366, A241918, A265146, A275024.
%K nonn,tabf
%O 2,2
%A _Franklin T. Adams-Watters_, Jan 22 2006
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