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A112798
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Table where n-th row is factorization of n, with each prime p_i replaced by i.
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1522
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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
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OFFSET
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2,2
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COMMENTS
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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
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
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)
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LINKS
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FORMULA
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EXAMPLE
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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
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MAPLE
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T:= n-> sort([seq(numtheory[pi](i[1])$i[2], i=ifactors(n)[2])])[]:
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.)
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MATHEMATICA
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PrimePi /@ Flatten[Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ Range@ 60 // Flatten // Rest (* Michael De Vlieger, May 09 2015 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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