login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A259254 Number of partitions of prime(n) into n primes. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

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).

LINKS

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

FORMULA

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

EXAMPLE

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].

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 *)

PROG

(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 "$_ ", A259254($_), "\n" for 1..60; # Dana Jacobsen, Dec 15 2015

CROSSREFS

Subsequence of A117278.

Cf. A000040.

Sequence in context: A002583 A068519 A108041 * A095017 A141559 A211395

Adjacent sequences:  A259251 A259252 A259253 * A259255 A259256 A259257

KEYWORD

nonn

AUTHOR

Doug Bell, Jun 22 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 02:42 EST 2020. Contains 338781 sequences. (Running on oeis4.)