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A121304 Number of parts in all the compositions of n into primes (i.e., in all ordered sequences of primes having sum n). 5
1, 1, 2, 5, 5, 14, 17, 32, 53, 76, 139, 198, 334, 515, 798, 1280, 1938, 3075, 4710, 7299, 11298, 17296, 26738, 40874, 62763, 96036, 146674, 224210, 341562, 520767, 792375, 1204951, 1831124, 2779234, 4217008, 6391663, 9683056, 14659038, 22177341 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

a(n) = Sum_{k=1..floor(n/2)} k*A121303(n,k).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 2..2000

FORMULA

G.f.: (Sum_{i>=1} z^prime(i))/(1 - Sum_{i>=1} z^prime(i))^2.

EXAMPLE

a(8) = 17 because the compositions of 8 into primes are [3,5], [5,3], [2,3,3], [3,2,3], [3,3,2] and [2,2,2,2], having a total of 2+2+3+3+3+4 = 17 parts.

MAPLE

g:=sum(z^ithprime(i), i=1..53)/(1-sum(z^ithprime(i), i=1..53))^2: gser:=series(g, z=0, 48): seq(coeff(gser, z, n), n=2..45);

# second Maple program:

b:= proc(n) option remember; `if`(n=0, [1, 0], add(

      `if`(isprime(j), (p->p+[0, p[1]])(b(n-j)), 0), j=1..n))

    end:

a:= n-> b(n)[2]:

seq(a(n), n=2..50);  # Alois P. Heinz, Nov 08 2013, revised Feb 12 2021

MATHEMATICA

nn=40; a[x_]:=Sum[x^Prime[n], {n, 1, nn}]; Drop[CoefficientList[Series[D[1/(1-y a[x]), y]/.y ->1, {x, 0, nn}], x], 2] (* Geoffrey Critzer, Nov 08 2013 *)

Table[Length[Flatten[Union[Flatten[Permutations/@Select[ IntegerPartitions[ n], AllTrue[ #, PrimeQ]&], 1]]]], {n, 2, 40}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 24 2016 *)

b[n_] := b[n] = If[n == 0, {1, 0}, Sum[If[PrimeQ[j],

     Function[p, p+{0, p[[1]]}][b[n-j]], {0, 0}], {j, 1, n}]];

a[n_] := b[n][[2]];

a /@ Range[2, 50] (* Jean-François Alcover, Jun 01 2021, after Alois P. Heinz *)

CROSSREFS

Cf. A023360, A121303.

Sequence in context: A154696 A154698 A063786 * A002106 A232316 A184604

Adjacent sequences:  A121301 A121302 A121303 * A121305 A121306 A121307

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Aug 06 2006

STATUS

approved

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Last modified June 24 05:58 EDT 2021. Contains 345416 sequences. (Running on oeis4.)