A259200 -id:A259200 - cp's OEIS frontend

A259201 Number of partitions of n into ten primes.

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 9, 11, 11, 14, 16, 18, 20, 25, 24, 31, 33, 38, 39, 48, 47, 59, 59, 69, 69, 87, 80, 102, 98, 118, 114, 143, 131, 168, 154, 191, 179, 227, 200, 261, 236, 297, 268, 344, 300, 396, 345, 442, 390, 509, 431, 576, 493, 641, 551, 729

Offset: 20

Doug Bell, Jun 20 2015

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

Alois P. Heinz, Table of n, a(n) for n = 20..10000
Index entries for sequences related to partitions

Column k=10 of A117278.
Number of partitions of n into r primes for r = 1-9: A010051, A061358, A068307, A259194, A259195, A259196, A259197, A259198, A259200.
Cf. A000040, A010051.

[#RestrictedPartitions(k,10,Set(PrimesUpTo(1000))):k in [20..80]] ; // Marius A. Burtea, Jul 13 2019

a(n) = [x^n y^10] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} A010051(r) * A010051(q) * A010051(p) * A010051(o) * A010051(m) * A010051(l) * A010051(k) * A010051(j) * A010051(i) * A010051(n-i-j-k-l-m-o-p-q-r). - Wesley Ivan Hurt, Jul 13 2019

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

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

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

A340965 Number of ways to write n as an ordered sum of 9 primes.

Original entry on oeis.org

1, 9, 36, 93, 198, 387, 696, 1170, 1845, 2740, 3960, 5562, 7566, 10125, 13248, 17133, 22014, 27774, 34776, 43173, 53010, 64869, 78696, 94617, 113415, 134946, 159552, 188164, 219960, 256041, 297180, 342846, 394614, 452595, 516276, 587997, 667938, 755109, 852444

Offset: 18

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 18..10000

Column k=9 of A121303.

Cf. A000040, A010051, A073610, A098238, A259200, A340960, A340961, A340962, A340963, A340964, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 9):
seq(a(n), n=18..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^9, {x, 0, nmax}], x] // Drop[#, 18] &

G.f.: (Sum_{k>=1} x^prime(k))^9.

A341457 Number of partitions of n into 9 nonprime parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 6, 7, 9, 10, 12, 15, 17, 20, 24, 28, 32, 38, 44, 51, 60, 67, 79, 91, 104, 120, 138, 154, 180, 203, 232, 262, 300, 335, 385, 428, 489, 543, 620, 688, 782, 861, 979, 1078, 1222, 1341, 1518, 1661, 1875, 2048, 2308

Offset: 9

Ilya Gutkovskiy, Feb 12 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..10000

Cf. A002095, A005171, A018252, A062610, A259200, A341408, A341451, A341452, A341453, A341454, A341455.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(isprime(i), 0, b(n-i, min(n-i, i), t-1))))
    end:
a:= n-> b(n$2, 9):
seq(a(n), n=9..68);  # Alois P. Heinz, Feb 12 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[PrimeQ[i], 0, b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 9];
Table[a[n], {n, 9, 68}] (* Jean-François Alcover, Feb 23 2022, after Alois P. Heinz *)

A326540 Sum of all the parts in the partitions of n into 9 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 19, 20, 42, 44, 69, 96, 100, 130, 189, 196, 261, 300, 341, 384, 528, 544, 700, 756, 888, 988, 1287, 1240, 1599, 1638, 2021, 2024, 2655, 2438, 3243, 3120, 3920, 3850, 4998, 4420, 6042, 5616, 7205, 6608

Offset: 0

Wesley Ivan Hurt, Jul 13 2019

nonn

Index entries for sequences related to partitions

Cf. A010051, A259200, A326541, A326542, A326543, A326544, A326545, A326546, A326547, A326548, A326549.

Table[n * Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (PrimePi[q] - PrimePi[q - 1]) (PrimePi[n - i - j - k - l - m - o - p - q] - PrimePi[n - i - j - k - l - m - o - p - q - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 50}]

a(n) = 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.
a(n) = n * A259200(n).
a(n) = A326541(n) + A326542(n) + A326543(n) + A326544(n) + A326545(n) + A326546(n) + A326547(n) + A326548(n) + A326549(n).

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A259201 Number of partitions of n into ten primes.

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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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A259198 Number of partitions of n into eight primes.

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A259194 Number of partitions of n into four primes.

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A259195 Number of partitions of n into five primes.

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A340965 Number of ways to write n as an ordered sum of 9 primes.