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

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

Author

Doug Bell, Jun 22 2015

Keywords

• nonn

Comments

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).

Examples

			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].
		

Links

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

Crossrefs

Subsequence of A117278.

Cf. A000040.

Programs

Maple

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

Mathematica

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

PARI

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

Perl

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

Perl

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

Formula

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