|
%I #105 Oct 21 2023 01:49:31
%S 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,
%T 36,40,48,64,17,26,33,35,42,44,45,50,54,56,60,72,80,96,128,19,34,39,
%U 49,52,55,63,66,70,75,81,84,88,90,100,108,112,120,144,160,192,256
%N 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).
%C The concatenation of all rows (with offset 1) gives a permutation of the natural numbers A000027 with fixed points 1-6, 9, 10, 14, 15, 21, 22, 33, 49, 1095199, ... and inverse permutation A215501.
%C 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.
%C The power prime(i)^j of the i-th prime is in row i*j for j in {0,1,2, ... }.
%C Column k=2 contains the even semiprimes A100484, where 10 and 22 are replaced by the odd semiprimes 9 and 21, respectively.
%C 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
%H Alois P. Heinz, <a href="/A215366/b215366.txt">Rows n = 0..26, flattened</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F 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
%e 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].
%e For n=0 the empty partition [] gives the empty product 1.
%e Triangle T(n,k) begins:
%e 1;
%e 2;
%e 3, 4;
%e 5, 6, 8;
%e 7, 9, 10, 12, 16;
%e 11, 14, 15, 18, 20, 24, 32;
%e 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64;
%e 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128;
%e ...
%e Corresponding triangle of integer partitions begins:
%e ();
%e 1;
%e 2, 11;
%e 3, 21, 111;
%e 4, 22, 31, 211, 1111;
%e 5, 41, 32, 221, 311, 2111, 11111;
%e 6, 42, 51, 33, 222, 411, 321, 2211, 3111, 21111, 111111;
%e 7, 61, 52, 43, 421, 511, 322, 331, 2221, 4111, 3211, 22111, 31111, 211111, 1111111; - _Gus Wiseman_, Dec 12 2016
%p b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
%p [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
%p end:
%p T:= n-> sort(b(n, n))[]:
%p seq(T(n), n=0..10);
%p # (2nd Maple program)
%p 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
%p # (3rd Maple program)
%p 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
%t 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_ *)
%t nn=7;HeinzPartition[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]//Reverse];
%t Take[GatherBy[Range[2^nn],Composition[Total,HeinzPartition]],nn+1] (* _Gus Wiseman_, Dec 12 2016 *)
%t Table[Map[Times @@ Prime@ # &, IntegerPartitions[n]], {n, 0, 8}] // Flatten (* _Michael De Vlieger_, Jul 12 2017 *)
%o (PARI) \\ From _M. F. Hasler_, Dec 06 2016 (Start)
%o A215366_row(n)=vecsort([vecprod([prime(p)|p<-P])|P<-partitions(n)]) \\ bug fix & syntax update by _M. F. Hasler_, Oct 20 2023
%o A215366_vec(N)=concat(apply(A215366_row,[0..N])) \\ "flattened" rows 0..N (End)
%Y Column k=1 gives: A008578(n+1).
%Y Last elements of rows give: A000079.
%Y Second to last elements of rows give: A007283(n-2) for n>1.
%Y Row sums give: A145519.
%Y Row lengths are: A000041.
%Y Cf. A129129 (with row elements using order of A080577).
%Y LCM of terms in row n gives A138534(n).
%Y Cf. A000027, A000040, A056239, A063008, A088850, A100484, A215501.
%Y Cf. A112798, A246867 (the same for partitions into distinct parts).
%Y Cf. A324939.
%K nonn,look,tabf
%O 0,2
%A _Alois P. Heinz_, Aug 08 2012
|
