A378175 Triangle T(n,k) read by rows in which n-th row lists in increasing order all multiplicative partitions mu of n (with factors > 1) encoded as Product_{j in mu} prime(j); n>=1, 1<=k<=A001055(n).
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 27, 23, 25, 29, 33, 31, 35, 37, 39, 45, 41, 43, 51, 47, 55, 49, 53, 57, 63, 81, 59, 61, 65, 69, 75, 67, 71, 77, 87, 99, 73, 85, 79, 93, 83, 89, 91, 95, 105, 111, 117, 135, 97, 121, 101, 123, 103, 115, 125, 107, 119, 129, 153
Offset: 1
Examples
The multiplicative partitions of n=8 are {[8], [4,2], [2,2,2]}, encodings give {prime(8), prime(4)*prime(2), prime(2)^3} = {19, 7*3, 3^3} => row 8 = [19, 21, 27].
For n=1 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
1 ;
3 ;
5 ;
7, 9 ;
11 ;
13, 15 ;
17 ;
19, 21, 27 ;
23, 25 ;
29, 33 ;
31 ;
35, 37, 39, 45 ;
41 ;
43, 51 ;
47, 55 ;
49, 53, 57, 63, 81 ;
59 ;
...
Links
- Alois P. Heinz, Rows n = 1..2047, flattened
Crossrefs
Programs
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Maple
b:= proc(n) option remember; `if`(n=1, {1}, {seq(map(x-> x* ithprime(d), b(n/d))[], d=numtheory[divisors](n) minus {1})}) end: T:= n-> sort([b(n)[]])[]: seq(T(n), n=1..28);