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 A215366 Triangle T(n,k) read by rows in which n-th row lists in increasing order all partitions lambda of n encoded as Product_{i in lambda} prime(i); n>=0, 1<=k<=A000041(n). 54
 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 14, 15, 18, 20, 24, 32, 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64, 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128, 19, 34, 39, 49, 52, 55, 63, 66, 70, 75, 81, 84, 88, 90, 100, 108, 112, 120, 144, 160, 192, 256 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The concatenation of all rows (with offset 1) gives a permutation of the natural numbers A000027 with fixed points 1-10, 14, 15, 21, 22, 33, 49, 1095199, ... and inverse permutation A215501. Number m is positioned in row n = A056239(m).  The number of different values m, such that both m and m+1 occur in row n is A088850(n).  A215369 lists all values m, such that both m and m+1 are in the same row. The power prime(i)^j of the i-th prime is in row i*j for j in {0,1,2, ... }. Column k=2 contains the even semiprimes A100484, where 10 and 22 are replaced by the odd semiprimes 9 and 21, respectively. This triangle is related to the triangle A145518, see in both triangles the first column, the right border, the second right border and the row sums. - Omar E. Pol, May 18 2015 LINKS Alois P. Heinz, Rows n = 0..26, flattened FORMULA Recurrence relation, explained for the set S(4) of entries in row 4: multiply the entries of S(3) by 2 (= 1st prime), multiply the entries of S(2) by 3 (= 2nd prime), multiply the entries of S(1) by 5 (= 3rd prime), multiply the entries of S(0) by 7 (= 4th prime); take the union of all the obtained products. The 3rd Maple program is based on this recurrence relation. - Emeric Deutsch, Jan 23 2016 EXAMPLE The partitions of n=3 are {[3], [2,1], [1,1,1]}, encodings give {prime(3), prime(2)*prime(1), prime(1)^3} = {5, 3*2, 2^3} => row 3 = [5, 6, 8]. For n=0 the empty partition [] gives the empty product 1. Triangle T(n,k) begins: :  1; :  2; :  3,  4; :  5,  6,  8; :  7,  9, 10, 12, 16; : 11, 14, 15, 18, 20, 24, 32; : 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64; : 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128; Corresponding triangle of integer partitions begins: (); 1; 2, 11; 3, 21, 111; 4, 22, 31, 211, 1111; 5, 41, 32, 221, 311, 2111, 11111; 6, 42, 51, 33, 222, 411, 321, 2211, 3111, 21111, 111111; 7, 61, 52, 43, 421, 511, 322, 331, 2221, 4111, 3211, 22111, 31111, 211111, 1111111;  - Gus Wiseman, Dec 12 2016 MAPLE b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],        [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])     end: T:= n-> sort(b(n, n))[]: seq(T(n), n=0..10); # (2nd Maple program) with(combinat): A := proc (n) local P, A, i: P := partition(n): A := {}; for i to nops(P) do A := `union`(A, {mul(ithprime(P[i][j]), j = 1 .. nops(P[i]))}) end do: A end proc; # the command A(m) yields row m. # Emeric Deutsch, Jan 23 2016 # (3rd Maple program) q:= 7: S[0] := {1}: for m to q do S[m] := `union`(seq(map(proc (f) options operator, arrow: ithprime(j)*f end proc, S[m-j]), j = 1 .. m)) end do; # for a given positive integer q, the program yields rows 0, 1, 2, ..., q. # Emeric Deutsch, Jan 23 2016 MATHEMATICA b[n_, i_] := b[n, i] = If[n == 0 || i<2, {2^n}, Table[Function[#*Prime[i]^j] /@ b[n - i*j, i-1], {j, 0, n/i}] // Flatten]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 12 2015, after Alois P. Heinz *) nn=7; HeinzPartition[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]//Reverse]; Take[GatherBy[Range[2^nn], Composition[Total, HeinzPartition]], nn+1] (* Gus Wiseman, Dec 12 2016 *) PROG (PARI) \\ From M. F. Hasler, Dec 06 2016 (Start) A215366_row(n)=apply(P->prod(i=1, #P, prime(P[i])), partitions(n)) A215366_vec(N)=concat(apply(A215366_row, [0..N])) \\ "flattened" rows 0..N (End) CROSSREFS Column k=1 gives: A008578(n+1). Last elements of rows give: A000079. Second to last elements of rows give: A007283(n-2) for n>1. Row sums give: A145519. Row lengths are: A000041. LCM of terms in row n gives A138534(n). Cf. A000027, A000040, A056239, A063008, A088850, A100484, A215501. Cf. A112798, A246867 (the same for partitions into distinct parts). Sequence in context: A117332 A242704 A243571 * A266195 A102530 A266196 Adjacent sequences:  A215363 A215364 A215365 * A215367 A215368 A215369 KEYWORD nonn,look,tabf AUTHOR Alois P. Heinz, Aug 08 2012 STATUS approved

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