A117278 -id:A117278 - cp's OEIS frontend

A259195 Number of partitions of n into five primes.

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

A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1

Offset: 0

Alois P. Heinz, Sep 28 2018

Equals A181506 when the first column is removed. - Georg Fischer, Jul 26 2023

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 0, 1;
  0, 0, 1, 0, 1;
  0, 0, 0, 1, 0, 1;
  0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A010054 (for n>0), A052344, A063993, A319814, A319815, A319816, A319817, A319818, A319819, A319820.
Row sums give A007294.
T(2n,n) gives A319799.
Cf. A000217, A117278, A181506, A243148, A319394.

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..20);

h[n_] := h[n] = If[n < 1, 0, If[IntegerQ @ Sqrt[8*n + 1], n, h[n - 1]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[ x*b[n - i, h[Min[n - i, i]]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, h[n]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000217(j)).

A344447 Number T(n,k) of partitions of n into k semiprimes; triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 0, 2, 1, 0, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 3, 2, 1, 0, 0, 1, 2, 1, 0, 0, 2, 3, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 1, 2, 3, 2, 1, 0, 0, 1, 3, 3, 1, 0, 0, 2, 3, 4, 2, 1, 1

Offset: 0

Alois P. Heinz, May 19 2021

T(n,k) is defined for all n,k >= 0. The triangle contains in each row n only the terms for k=0 and then up to the last positive T(n,k) (if it exists).

			Triangle T(n,k) begins:
  1 ;
  0 ;
  0 ;
  0 ;
  0, 1 ;
  0    ;
  0, 1 ;
  0    ;
  0, 0, 1 ;
  0, 1    ;
  0, 1, 1 ;
  0       ;
  0, 0, 1, 1 ;
  0, 0, 1    ;
  0, 1, 1, 1 ;
  0, 1, 1    ;
  0, 0, 1, 1, 1 ;
  0, 0, 0, 1    ;
  0, 0, 2, 2, 1 ;
  0, 0, 2, 1    ;
  0, 0, 2, 1, 1, 1 ;
  ...

Alois P. Heinz, Rows n = 0..500, flattened

Columns k=0-10 give: A000007, A064911, A072931, A344446, A340756, A344245, A344246, A344254, A344255, A344256, A344257.
Row sums give A101048.
T(4n,n) gives A000012.
Cf. A001358, A117278, A281617.

h:= proc(n) option remember; `if`(n=0, 0,
     `if`(numtheory[bigomega](n)=2, n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
     `if`(i>n, 0, expand(x*b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, h(n))):
seq(T(n), n=0..32);

h[n_] := h[n] = If[n == 0, 0,
     If[PrimeOmega[n] == 2, n, h[n-1]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     If[i > n, 0, Expand[x*b[n-i, h[Min[n-i, i]]]]] + b[n, h[i-1]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, Max[0, Exponent[#, x]]}]&[b[n, h[n]]];
Table[T[n], {n, 0, 32}] // Flatten (* Jean-François Alcover, Aug 19 2021, after Alois P. Heinz *)

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A001358(j)).

Sum_{k>0} k * T(n,k) = A281617(n).

A259254 Number of partitions of prime(n) into n primes.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 2, 2, 3, 7, 7, 12, 19, 19, 25, 44, 72, 72, 119, 147, 152, 234, 292, 435, 777, 920, 946, 1135, 1161, 1377, 3703, 4294, 5944, 5944, 10742, 10742, 14488, 18958, 22092, 28662, 37687, 37687, 63068, 63068, 72400, 72400, 132756, 233796, 265315, 265315

Offset: 1

Doug Bell, Jun 22 2015

nonn

a(n) = number of partitions of A000040(n) into n primes.

If n > 1 and prime(n) - prime(n-1) = 2 (twin primes), then the number of partitions of prime(n) into n primes that don't contain 2 is equal to a(n) - a(n-1); every partition of primes in a(n) that does contain a 2 matches a partition of primes in a(n-1) with an added partition for 2. Further, if n is even, then a(n) = a(n-1).

			a(9) = 3 because 23 is the ninth prime number (A000040(9) = 23), and 23 can be partitioned into nine primes in three ways: [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].

Alois P. Heinz, Table of n, a(n) for n = 1..600 (first 100 terms from Doug Bell)

Subsequence of A117278.

Cf. A000040.

N:= 100:  # to get a(1) to a(N)
Primes:= [seq(ithprime(i),i=1..N)]:
W:= proc(n,m,j) option remember;
  if n < 0 then return 0 fi;
  if n=0 then if m=0 then return 1 else return 0 fi fi;
  add(W(n-Primes[i],m-1,i),i=1..j)
end proc:
seq(W(Primes[n],n,n), n = 1 .. N); # Robert Israel, Jun 22 2015

f[n_] := Length@ IntegerPartitions[ Prime@n, {n}, Prime@ Range@ n]; Array[f, 50] (* Giovanni Resta, Jun 23 2015 *)

a(n) = {nb = 0; forpart(p=prime(n), ok=1; for (k=1, n, if (!isprime(p[k]), ok=0; break));nb += ok,[2,prime(n)],[n,n]); nb;} \\ Michel Marcus, Jun 23 2015

use ntheory ":all"; use List::MoreUtils qw/all/; sub a259254 { my($n,$sum)=(shift,0); forpart { $sum++ if all { is_prime($) } @; } nth_prime($n),{n=>$n,amin=>2}; $sum; } say a259254($) for 1..60; # _Dana Jacobsen, Dec 15 2015

use ntheory ":all";
use Memoize;  memoize 'W';
sub W {
  my($n, $m, $j) = @_;
  return 0 if $n < 0;
  return ($m == 0) ? 1 : 0  if $n == 0;
  vecsum( map { W($n-nth_prime($), $m-1, $) } 1 .. $j );
}
sub A259254 { my $n = shift; W(nth_prime($n), $n, $n); }
print "$A259254($"> ", A259254($),"\n" for 1..60; # Dana Jacobsen, Dec 15 2015

a(n) = A117278(A000040(n),n). - Robert Israel, Jun 22 2015

A347578 Number of partitions of n into at most 4 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 10, 9, 11, 11, 13, 11, 15, 12, 16, 15, 16, 15, 18, 16, 20, 17, 23, 18, 24, 20, 26, 22, 26, 23, 31, 23, 33, 26, 35, 26, 39, 27, 41, 32, 41, 31, 46, 31, 48, 34, 51, 34, 54, 36, 58, 40, 58, 42, 64, 41

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Cf. A000040, A000607, A071335, A117278, A259194, A347552, A347609, A347610.

a(n) = Sum_{k=1..4} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021

A347609 Number of partitions of n into at most 5 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 8, 9, 10, 11, 11, 14, 14, 15, 17, 18, 19, 21, 22, 23, 25, 27, 27, 32, 29, 34, 33, 37, 37, 42, 39, 47, 44, 51, 47, 58, 50, 61, 57, 67, 61, 73, 65, 80, 71, 86, 75, 95, 79, 101, 86, 107, 92, 115, 95, 125, 103, 132, 108

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Cf. A000040, A000607, A071335, A117278, A259195, A347552, A347578, A347610.

a(n) = Sum_{k=1..5} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021

A347610 Number of partitions of n into at most 6 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 9, 11, 12, 14, 15, 17, 18, 20, 23, 24, 27, 28, 32, 33, 37, 38, 43, 44, 48, 51, 55, 55, 63, 62, 70, 71, 77, 78, 89, 87, 97, 96, 108, 105, 121, 114, 133, 127, 144, 139, 161, 149, 174, 165, 189, 177, 208, 188, 226, 206

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Cf. A000040, A000607, A071335, A117278, A259196, A347552, A347578, A347609.

a(n) = Sum_{k=1..6} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021

A299168 Number of ordered ways of writing n-th prime number as a sum of n primes.

Original entry on oeis.org

1, 0, 0, 0, 5, 6, 42, 64, 387, 5480, 10461, 113256, 507390, 1071084, 4882635, 44984560, 382362589, 891350154, 7469477771, 33066211100, 78673599501, 649785780710, 2884039365010, 22986956007816, 306912836483025, 1361558306986280, 3519406658042964

Offset: 1

Ilya Gutkovskiy, Feb 04 2018

nonn

			a(5) = 5 because fifth prime number is 11 and we have [3, 2, 2, 2, 2], [2, 3, 2, 2, 2], [2, 2, 3, 2, 2], [2, 2, 2, 3, 2] and [2, 2, 2, 2, 3].

Alois P. Heinz, Table of n, a(n) for n = 1..400

Cf. A000040, A000586, A000607, A023360, A056768, A070215, A073610, A098238, A117278, A121303, A219180, A224344, A259254, A265112, A341459.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(isprime(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(ithprime(n), n):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 13 2021

Table[SeriesCoefficient[Sum[x^Prime[k], {k, 1, n}]^n, {x, 0, Prime[n]}], {n, 1, 27}]

a(n) = [x^prime(n)] (Sum_{k>=1} x^prime(k))^n.

A347552 Number of partitions of n into at most 2 prime parts.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Cf. A000040, A000607, A061358, A071335, A117278, A347578, A347609, A347610.

a(n) = Sum_{k=1..2} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021

A301971 a(n) = [x^n] Product_{k>=1} 1/(1 - x^prime(k))^n.

Original entry on oeis.org

1, 0, 2, 3, 10, 30, 77, 252, 682, 2136, 6182, 18766, 56173, 169351, 512990, 1551828, 4720170, 14348289, 43751984, 133502873, 408029510, 1248460587, 3823949824, 11724787763, 35980251181, 110510334780, 339674840715, 1044812449722, 3215861978150, 9904301974294, 30521063942312, 94103983534015

Offset: 0

Ilya Gutkovskiy, Mar 29 2018

nonn

Number of partitions of n into prime parts of n kinds.

Index entries for sequences related to partitions

Cf. A000607, A008485, A030009, A117278, A219224, A259254, A291647, A298436, A299168.

Table[SeriesCoefficient[Product[1/(1 - x^Prime[k])^n, {k, 1, n}], {x, 0, n}], {n, 0, 31}]

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A259195 Number of partitions of n into five primes.

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A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A344447 Number T(n,k) of partitions of n into k semiprimes; triangle T(n,k), n>=0, read by rows.

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A259254 Number of partitions of prime(n) into n primes.

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A347578 Number of partitions of n into at most 4 prime parts.

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A347609 Number of partitions of n into at most 5 prime parts.

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A347610 Number of partitions of n into at most 6 prime parts.

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A299168 Number of ordered ways of writing n-th prime number as a sum of n primes.

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