login
This site is supported by donations to The OEIS Foundation.

 

Logo

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A219180 Number T(n,k) of partitions of n into k distinct prime parts; triangle T(n,k), n>=0, read by rows. 18
1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 1, 1, 1, 0, 0, 2, 2, 0, 0, 1, 2, 1, 0, 0, 2, 2, 0, 1, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 2, 2, 0, 0, 2, 3, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,41

COMMENTS

T(n,k) is defined for all n>=0 and k>=0.  The triangle contains only elements with 0 <= k <= A024936(n).  T(n,k) = 0 for k > A024936(n).  Three rows are empty because there no partitions of n into distinct prime parts for n in {1,4,6}.

LINKS

Alois P. Heinz, Rows n = 0..1000, flattened

FORMULA

G.f. of column k: Sum_{0<i_1<i_2<...<i_k} x^(Sum_{j=1..k} prime(i_j)).

T(n,k) = [x^n*y^k] Product_{i>=1} (1+x^prime(i)*y).

EXAMPLE

T(0,0) = 1: [], the empty partition.

T(2,1) = 1: [2].

T(5,1) = 1: [5], T(5,2) = 1: [2,3].

T(16,2) = 2: [5,11], [3,13].

Triangle T(n,k) begins:

1;

;

0, 1;

0, 1;

;

0, 1, 1;

;

0, 1, 1;

0, 0, 1;

0, 0, 1;

0, 0, 1, 1;

0, 1;

0, 0, 1, 1;

MAPLE

b:= proc(n, i) option remember;

      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),

       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))

    end:

T:= proc(n) local l; l:= b(n, numtheory[pi](n));

       while nops(l)>0 and l[-1]=0 do l:= subsop(-1=NULL, l) od; l[]

    end:

seq(T(n), n=0..50);

MATHEMATICA

nn=20; a=Table[Prime[n], {n, 1, nn}]; CoefficientList[Series[Product[1+y x^a[[i]], {i, 1, nn}], {x, 0, nn}], {x, y}]//Grid  (* Geoffrey Critzer, Nov 21 2012 *)

zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]]; b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, zip[Plus, b[n, i-1], Join[{0}, If[Prime[i] > n, {}, b[n-Prime[i], i-1]]], 0]]]; T[n_] := Module[{l}, l = b[n, PrimePi[n]]; While[Length[l]>0 && l[[-1]] == 0, l = ReplacePart[l, -1 -> Sequence[]]]; l]; Table[T[n], {n, 0, 50}] // Flatten (* Jean-Fran├žois Alcover, Jan 29 2014, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000007, A010051, A117929, A125688, A219198, A219199, A219200, A219201, A219202, A219203, A219204.

Row lengths are 1 + A024936(n).

Row sums give: A000586.

Last elements of rows give: A219181.

Row maxima give: A219182.

Least n with T(n,k) > 0 is A007504(k).

Cf. A000040, A117278, A219107.

Sequence in context: A085252 A250214 A073423 * A179952 A134023 A257931

Adjacent sequences:  A219177 A219178 A219179 * A219181 A219182 A219183

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Nov 13 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 18 14:09 EDT 2017. Contains 290720 sequences.