A061358 -id:A061358 - cp's OEIS frontend

A259200 Number of partitions of n into nine primes.

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 7, 9, 10, 11, 12, 16, 16, 20, 21, 24, 26, 33, 31, 39, 39, 47, 46, 59, 53, 69, 65, 80, 77, 98, 85, 114, 104, 131, 118, 154, 133, 179, 155, 200, 177, 236, 196, 268, 227, 300, 256

Offset: 18

Doug Bell, Jun 20 2015

			a(23) = 3 because there are 3 partitions of 23 into nine primes: [2,2,2,2,2,2,2,2,7], [2,2,2,2,2,2,3,3,5] and [2,2,2,2,3,3,3,3,3].

Robert Israel, Table of n, a(n) for n = 18..10000
Index entries for sequences related to partitions

Column k=9 of A117278.
Number of partitions of n into r primes for r = 1..10: A010051, A061358, A068307, A259194, A259195, A259196, A259197, A259198, this sequence, A259201.
Cf. A000040.

[#RestrictedPartitions(k,9,Set(PrimesUpTo(1000))):k in [18..70]] ; // Marius A. Burtea, Jul 13 2019

N:= 100: # to get a(0) to a(N)
Primes:= select(isprime,[$1..N]):
np:= nops(Primes):
for j from 0 to np do g[0,j]:= 1 od:
for n from 1 to 9 do
  g[n,0]:= 0:
  for j from 1 to np do
     g[n,j]:= convert(series(add(g[k,j-1]
          *x^((n-k)*Primes[j]),k=0..n),x,N+1),polynom)
  od
od:
seq(coeff(g[9,np],x,i),i=18..N) # Robert Israel, Jun 21 2015

Table[Length[Select[IntegerPartitions[n],Length[#]==9&&AllTrue[ #, PrimeQ]&]], {n,18,70}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 31 2016 *)

a(n) = {nb = 0; forpart(p=n, if (#p && (#select(x->isprime(x), Vec(p)) == #p), nb+=1), , [9,9]); nb;} \\ Michel Marcus, Jun 21 2015

a(n) = [x^n y^9] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q), where c = A010051. - Wesley Ivan Hurt, Jul 13 2019

A062610 Number of ways of writing n = c1 + c2 with c1 and c2 nonprimes [=1 or composite].

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 4, 3, 5, 3, 6, 4, 6, 4, 7, 5, 8, 6, 7, 6, 9, 6, 11, 7, 8, 8, 11, 8, 12, 9, 11, 9, 12, 9, 14, 10, 14, 10, 14, 11, 16, 12, 15, 12, 16, 13, 19, 14, 15, 14, 19, 14, 21, 15, 17, 16, 21, 16, 22, 17, 20, 17, 22, 17, 25, 18, 22, 19, 23, 19

Offset: 1

Labos Elemer, Jul 04 2001

			n = 22 has floor(n/2) = 11 partitions of form n = a+b; 3 partitions are of prime+prime [3+19 = 5+17 = 11+11], 3 partitions are of prime+nonprime [2+20 = 7+15 = 13+9], 5 partitions are of nonprime+nonprime [1+21 = 4+18 = 6+16 = 8+14 = 10+12]. So a(22) = 5.

Cf. A010051, A014092, A061358, A062602.

Table[Count[Transpose@ {#, n - #} &@ Range[Floor[n/2]], w_ /; Times @@ Boole@ Map[! PrimeQ@ # &, w] == 1], {n, 83}] (* Michael De Vlieger, Jul 04 2016 *)

a(n) = Sum_{i=1..floor(n/2)} (1 - A010051(i)) * (1 - A010051(n-i)) = Sum_{i=1..floor(n/2)} A005171(i)*A005171(n-i). - Wesley Ivan Hurt, Apr 08 2018

a(n) + A061358(n) + A062602(n) = A004526(n). - R. J. Mathar, Sep 10 2021

A117278 Triangle read by rows: T(n,k) is the number of partitions of n into k prime parts (n>=2, 1<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 1, 1, 0, 2, 2, 1, 0, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 0, 2, 1, 3, 2, 1, 1, 0, 1, 3, 2, 3, 2, 1, 0, 2, 2, 3, 3, 2, 1, 1, 1, 0, 4, 3, 3, 3, 2, 1, 0, 2, 2, 4, 3, 4, 2, 1, 1, 1, 1, 3, 4, 5, 3, 3, 2, 1, 0, 2, 2, 6, 4, 4, 4, 2, 1, 1, 0, 1, 5, 3, 6

Offset: 2

Emeric Deutsch, Mar 07 2006

Row n has floor(n/2) terms. Row sums yield A000607. T(n,1) = A010051(n) (the characteristic function of the primes). T(n,2) = A061358(n). Sum(k*T(n,k), k>=1) = A084993(n).

			T(12,3) = 2 because we have [7,3,2] and [5,5,2].
Triangle starts:
  1;
  1;
  0, 1;
  1, 1;
  0, 1, 1;
  1, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1;
  ...

Alois P. Heinz, Rows n = 2..200, flattened

Columns k=1-10 give: A010051, A061358, A068307, A259194, A259195, A259196, A259197, A259198, A259200, A259201.
Row sums give A000607.
T(A000040(n),n) gives A259254(n).
Cf. A084993, A219180.

g:=1/product(1-t*x^(ithprime(j)),j=1..30): gser:=simplify(series(g,x=0,30)): for n from 2 to 22 do P[n]:=sort(coeff(gser,x^n)) od: for n from 2 to 22 do seq(coeff(P[n],t^j),j=1..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
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))[]], 0)))
    end:
T:= n-> subsop(1=NULL, b(n, numtheory[pi](n)))[]:
seq(T(n), n=2..25);  # Alois P. Heinz, Nov 16 2012

(* As triangle: *) nn=20;a=Product[1/(1-y x^i),{i,Table[Prime[n],{n,1,nn}]}];Drop[CoefficientList[Series[a,{x,0,nn}],{x,y}],2,1]//Grid (* Geoffrey Critzer, Oct 30 2012 *)

parts(n, pred)={prod(k=1, n, if(pred(k), 1/(1-y*x^k) + O(x*x^n), 1))}
{my(n=15); apply(p->Vecrev(p/y), Vec(parts(n, isprime)-1))} \\ Andrew Howroyd, Dec 28 2017

G.f.: G(t,x) = -1+1/product(1-tx^(p(j)), j=1..infinity), where p(j) is the j-th prime.

A259196 Number of partitions of n into six primes.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 3, 4, 5, 6, 6, 8, 7, 10, 10, 12, 11, 16, 12, 19, 17, 22, 18, 26, 20, 31, 24, 33, 27, 42, 29, 47, 35, 51, 38, 60, 41, 68, 47, 73, 53, 86, 54, 95, 64, 103, 70, 116, 73, 131, 81, 137, 89, 156, 92, 171, 103, 180, 112, 202, 117, 223, 127, 232

Offset: 12

Doug Bell, Jun 20 2015

			a(17) = 3 because there are 3 partitions of 17 into six primes: [2,2,2,2,2,7], [2,2,2,3,3,5] and [2,3,3,3,3,3].

Alois P. Heinz, Table of n, a(n) for n = 12..10000
Index entries for sequences related to partitions

Column k=6 of A117278.
Number of partitions of n into r primes for r = 1-10: A010051, A061358, A068307, A259194, A259195, this sequence, A259197, A259198, A259200, A259201.
Cf. A000040.

Table[Count[IntegerPartitions[n,{6}],?(AllTrue[#,PrimeQ]&)],{n,12,80}] (* _Harvey P. Dale, Jul 27 2024 *)

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} A010051(i) * A010051(j) * A010051(k) * A010051(l) * A010051(m) * A010051(n-i-j-k-l-m). - Wesley Ivan Hurt, Apr 17 2019

a(n) = [x^n y^6] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

A259197 Number of partitions of n into seven primes.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 4, 6, 6, 8, 8, 9, 10, 14, 12, 16, 16, 19, 19, 26, 22, 30, 26, 34, 31, 43, 33, 48, 42, 56, 47, 66, 51, 77, 60, 84, 68, 99, 73, 112, 86, 123, 95, 143, 103, 162, 116, 174, 131, 200, 137, 220, 156, 241, 171, 270, 180, 300, 202, 322, 223, 359

Offset: 14

Doug Bell, Jun 20 2015

			a(17) = 2 because there are 2 partitions of 17 into seven primes: [2,2,2,2,2,2,5] and [2,2,2,2,3,3,3].

Alois P. Heinz, Table of n, a(n) for n = 14..10000
Index entries for sequences related to partitions

Column k=7 of A117278.
Number of partitions of n into r primes for r = 1-10: A010051, A061358, A068307, A259194, A259195, A259196, this sequence, A259198, A259200, A259201.
Cf. A000040.

Table[Length@IntegerPartitions[n, {7}, Prime@Range@100], {n, 14, 100}] (* Robert Price, Apr 25 2025 *)

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} A010051(i) * A010051(j) * A010051(k) * A010051(l) * A010051(m) * A010051(o) * A010051(n-i-j-k-l-m-o). - Wesley Ivan Hurt, Apr 17 2019

a(n) = [x^n y^7] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

A259198 Number of partitions of n into eight primes.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10, 9, 12, 14, 16, 16, 21, 19, 26, 26, 31, 30, 39, 34, 46, 43, 53, 48, 65, 56, 77, 66, 85, 77, 104, 84, 118, 99, 133, 112, 155, 123, 177, 143, 196, 162, 227, 174, 256, 200, 282, 220, 318, 241, 360, 270, 389, 300, 442, 322

Offset: 16

Doug Bell, Jun 20 2015

			a(20) = 2 because there are 2 partitions of 20 into eight primes: [2,2,2,2,2,2,3,5] and [2,2,2,2,3,3,3,3].

Alois P. Heinz, Table of n, a(n) for n = 16..10000
Index entries for sequences related to partitions

Column k=8 of A117278.
Number of partitions of n into r primes for r = 1-10: A010051, A061358, A068307, A259194, A259195, A259196, A259197, this sequence, A259200, A259201.
Cf. A000040.

Table[Length@IntegerPartitions[n, {8}, Prime@Range@100], {n, 16, 100}] (* Robert Price, Apr 25 2025 *)

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} A010051(i) * A010051(j) * A010051(k) * A010051(l) * A010051(m) * A010051(o) * A010051(p) * A010051(n-i-j-k-l-m-o-p). - Wesley Ivan Hurt, Apr 17 2019
a(n) = [x^n y^8] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019
a(n) = A326455(n)/n for n > 0. - Wesley Ivan Hurt, Jul 06 2019

A117929 Number of partitions of n into 2 distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 2, 0, 3, 1, 2, 0, 2, 0, 3, 1, 2, 1, 3, 0, 4, 0, 1, 1, 3, 0, 4, 1, 3, 1, 3, 0, 5, 1, 4, 0, 3, 0, 5, 1, 3, 0, 3, 0, 6, 1, 2, 1, 5, 0, 6, 0, 2, 1, 5, 0, 6, 1, 4, 1, 5, 0, 7, 0, 4, 1, 4, 0, 8, 1, 4, 0, 4, 0, 9, 1, 4, 0, 4, 0, 7, 0, 3, 1, 6, 0, 8, 1, 5, 1

Offset: 1

Emeric Deutsch, Apr 03 2006

Number of distinct rectangles with prime length and width such that L + W = n, W < L. For example, a(16) = 2; the two rectangles are 3 X 13 and 5 X 11. - Wesley Ivan Hurt, Oct 29 2017

			a(24) = 3 because we have [19,5], [17,7] and [13,11].

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A010051, A045917, A061358, A073610, A166081 (positions of 0), A077914 (positions of 2), A080862 (positions of 6).

Column k=2 of A219180. - Alois P. Heinz, Nov 13 2012

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j-1),j=1..35): gser:=series(g,x=0,130): seq(coeff(gser,x,n),n=1..125);
# alternative
A117929 := proc(n)
    local a,i,p ;
    a := 0 ;
    p := 2 ;
    for i from 1 do
        if 2*p >= n then
            return a;
        end if;
        if isprime(n-p) then
            a := a+1 ;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
seq(A117929(n),n=1..80) ; # R. J. Mathar, Oct 01 2021

l = {}; For[n = 1, n <= 1000, n++, c = 0; For[k = 1, Prime[k] < n/2, k++, If[PrimeQ[n - Prime[k]], c = c + 1] ]; AppendTo[l, c] ] l (* Jake Foster, Oct 27 2008 *)
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[#,PrimeQ]&&#[[1]]!= #[[2]] &)],{n,120}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 26 2020 *)

a(n)=my(s);forprime(p=2,(n-1)\2,s+=isprime(n-p));s \\ Charles R Greathouse IV, Feb 26 2014

from sympy import sieve
from collections import Counter
from itertools import combinations
def aupton(max):
    sieve.extend(max)
    a = Counter(c[0]+c[1] for c in combinations(sieve._list, 2))
    return [a[n] for n in range(1, max+1)]
print(aupton(105)) # Michael S. Branicky, Feb 16 2024

G.f.: Sum_{j>0} Sum_{i=1..j-1} x^(p(i)+p(j)), where p(k) is the k-th prime.
G.f.: A(x)^2/2 - A(x^2)/2 where A(x) = Sum_{p in primes} x^p. - Geoffrey Critzer, Nov 21 2012
a(n) = [x^n*y^2] Product_{i>=1} (1+x^prime(i)*y). - Alois P. Heinz, Nov 22 2012
a(n) = Sum_{i=2..floor((n-1)/2)} A010051(i) * A010051(n-i). - Wesley Ivan Hurt, Oct 29 2017

A259194 Number of partitions of n into four primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 2, 3, 3, 4, 4, 6, 3, 6, 5, 7, 5, 9, 5, 11, 7, 11, 7, 13, 6, 14, 9, 15, 8, 18, 9, 21, 10, 19, 11, 24, 10, 26, 12, 26, 13, 30, 12, 34, 15, 33, 16, 38, 14, 41, 17, 41, 16, 45, 16, 50, 19, 47, 21, 56, 20, 61, 20, 57

Offset: 0

Doug Bell, Jun 20 2015

			a(17) = 3 because 17 can be written as the sum of four primes in exactly three ways: 2+2+2+11, 2+3+5+7 and 2+5+5+5.

Giovanni Resta, Table of n, a(n) for n = 0..5000
Index entries for sequences related to partitions

Column k=4 of A117278.
Number of partitions of n into r primes for r = 1..10: A010051, A061358, A068307, this sequence, A259195, A259196, A259197, A259198, A259200, A259201.
Cf. A000040.

[0] cat [#RestrictedPartitions(n,4,{d:d in PrimesUpTo(n)}):n in [1..100]]; // Marius A. Burtea, May 07 2019

a[n_] := Length@ IntegerPartitions[n, {4}, Prime@ Range@ PrimePi@ n]; a /@
Range[0, 100] (* Giovanni Resta, Jun 21 2015 *)
Table[Count[IntegerPartitions[n,{4}],?(AllTrue[#,PrimeQ]&)],{n,0,80}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Feb 03 2019 *)

a(n) = {nb = 0; forpart(p=n, if (#p && (#select(x->isprime(x), Vec(p)) == #p), nb+=1), , [4,4]); nb;} \\ Michel Marcus, Jun 21 2015

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} A010051(i) * A010051(j) * A010051(k) * A010051(n-i-j-k). - Wesley Ivan Hurt, Apr 17 2019

a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

A051034 Minimal number of primes needed to sum to n.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2

Offset: 2

Eric W. Weisstein

			a(2) = 1 because 2 is already prime.
a(4) = 2 because 4 = 2+2 is a partition of 4 into 2 prime parts and there is no such partition with fewer terms.
a(27) = 3 because 27 = 3+5+19 is a partition of 27 into 3 prime parts and there is no such partition with fewer terms.

T. D. Noe, Table of n, a(n) for n = 2..10000
Olivier Ramaré, On Šnirel'man's constant, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, 22:4 (1995), pp. 645-706.
Yannick Saouter, Vinogradov's theorem is true up to 10^20
Terence Tao, Every odd number greater than 1 is the sum of at most five primes, arXiv:1201.6656 [math.NT], 2012 preprint, to appear in Mathematics of Computation.
Eric Weisstein's World of Mathematics, Prime Partition
Index entries for sequences related to Goldbach conjecture

Cf. A025583, A004526, A007944, A007962, A000607, A051036, A010051, A061358, A068307, A103765.

Different from A072491.

(* Assuming Goldbach's conjecture *) a[p_?PrimeQ] = 1; a[n_] := If[ Reduce[ n == x + y, {x, y}, Primes] === False, 3, 2]; Table[a[n], {n, 2, 112}] (* Jean-François Alcover, Apr 03 2012 *)

issum(n,k)=if(k==1,isprime(n),k--;forprime(p=2,n,if(issum(n-p,k),return(1)));0)
a(n)=my(k);while(!issum(n,k++),);k \\ Charles R Greathouse IV, Jun 01 2011

a(n) = 1 iff n is prime. a(2n) = 2 (for n > 1) if Goldbach's conjecture is true. a(2n+1) = 2 (for n >= 1) if 2n+1 is not prime, but 2n-1 is. a(2n+1) >= 3 (for n >= 1) if both 2n+1 and 2n-1 are not primes (for sufficiently large n, a(2n+1) = 3 by Vinogradov's theorem, 1937). - Franz Vrabec, Nov 30 2004
a(n) <= 3 for all n, assuming the Goldbach conjecture. - N. J. A. Sloane, Jan 20 2007
a(2n+1) <= 5, see Tao 2012. - Charles R Greathouse IV, Jul 09 2012
Assuming Goldbach's conjecture, a(n) <= 3. In particular, a(p)=1; a(2*n)=2 for n>1; a(p+2)=2 provided p+2 is not prime; otherwise a(n)=3. - Sean A. Irvine, Jul 29 2019
a(2n+1) <= 3 by Helfgott's proof of Goldbach's ternary conjecture, and hence a(n) <= 4 in general. - Charles R Greathouse IV, Oct 24 2022

More terms from Naohiro Nomoto, Mar 16 2001

A259195 Number of partitions of n into five primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 5, 4, 6, 6, 7, 6, 10, 7, 11, 9, 12, 11, 17, 11, 18, 13, 20, 14, 24, 15, 27, 18, 29, 21, 35, 19, 38, 24, 41, 26, 47, 26, 53, 30, 54, 34, 64, 33, 70, 38, 73, 41, 81, 41, 89, 45, 92, 50, 103, 47, 112, 56, 117, 61, 127, 57

Offset: 0

Doug Bell, Jun 20 2015

			a(17) = 3 because 17 can be written as the sum of five primes in exactly three ways: 2+2+3+3+7, 2+2+3+5+5, and 3+3+3+3+5.

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 5001 terms from Doug Bell)
Index entries for sequences related to partitions
Sean A. Irvine, Java program (github)

Column k=5 of A117278.
Number of partitions of n into r primes for r = 1..10: A010051, A061358, A068307, A259194, this sequence, A259196, A259197, A259198, A259200, A259201.
Cf. A000040.

[0] cat [#RestrictedPartitions(n,5,{p:p in PrimesUpTo(n)}):n in [1..70]]; // Marius A. Burtea, May 09 2019

Array[Count[IntegerPartitions[#, {5}], ?(AllTrue[#, PrimeQ] &)] &, 71] (* _Michael De Vlieger, Apr 21 2019 *)

a(n) = {nb = 0; forpart(p=n, if (#p && (#select(x->isprime(x), Vec(p)) == #p), nb+=1), , [5,5]); nb;} \\ Michel Marcus, Jun 21 2015

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} c(i) * c(j) * c(k) * c(l) * c(n-i-j-k-l), where c = A010051. - Wesley Ivan Hurt, Apr 17 2019

a(n) = [x^n y^5] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

More terms from David A. Corneth, Sep 06 2020

cp's OEIS Frontend

A259200 Number of partitions of n into nine primes.

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A062610 Number of ways of writing n = c1 + c2 with c1 and c2 nonprimes [=1 or composite].

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A117278 Triangle read by rows: T(n,k) is the number of partitions of n into k prime parts (n>=2, 1<=k<=floor(n/2)).

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A259196 Number of partitions of n into six primes.

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A259197 Number of partitions of n into seven primes.

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Mathematica

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A259198 Number of partitions of n into eight primes.

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Mathematica

Formula

A117929 Number of partitions of n into 2 distinct primes.

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Maple

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PARI

Python