A056768 Number of partitions of the n-th prime into parts that are all primes.

1, 1, 2, 3, 6, 9, 17, 23, 40, 87, 111, 219, 336, 413, 614, 1083, 1850, 2198, 3630, 5007, 5861, 9282, 12488, 19232, 33439, 43709, 49871, 64671, 73506, 94625, 221265, 279516, 394170, 441250, 766262, 853692, 1175344, 1608014, 1975108, 2675925

Offset

1,3

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 4000 terms from Alois P. Heinz)

Formula

a(n) = A000607(prime(n)).

a(n) = A168470(n) + 1. - Alonso del Arte, Feb 15 2014, restating the corresponding formula given by R. J. Mathar for A168470.

a(n) = [x^prime(n)] Product_{k>=1} 1/(1 - x^prime(k)). - Ilya Gutkovskiy, Jun 05 2017

Example

a(4) = 3 because the 4th prime is 7 which can be partitioned using primes in 3 ways: 7, 5 + 2, and 3 + 2 + 2.
In connection with the 6th prime 13, for instance, we have the a(6) = 9 prime partitions: 13 = 2 + 2 + 2 + 2 + 2 + 3 = 2 + 2 + 2 + 2 + 5 = 2 + 2 + 2 + 7 = 2 + 2 + 3 + 3 + 3 = 2 + 3 + 3 + 5 = 2 + 11 = 3 + 3 + 7 = 3 + 5 + 5.

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=2
       and n::even, 1, `if`(i=2 or n=1, 0,
       b(n, prevprime(i)))+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> b(ithprime(n)$2):
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 15 2016

Mathematica

Table[Count[IntegerPartitions[n], _?(AllTrue[#, PrimeQ]&)], {n, Prime[ Range[ 40]]}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 07 2015 *)
n=40; ser=Product[1/(1-x^Prime[i]), {i, 1, n}]; Table[SeriesCoefficient[ser, {x, 0, Prime[i]}], {i, 1, n}] (* Gus Wiseman, Sep 14 2016 *)

Prog

(Haskell)
a056768 = a000607 . a000040  -- Reinhard Zumkeller, Aug 05 2012

Crossrefs

Cf. A000041, A000607, A100118, A276687, A070215 (distinct parts).

Sequence in context: A018721 A018384 A282842 * A029511 A320271 A056532

Adjacent sequences: A056765 A056766 A056767 * A056769 A056770 A056771

Keyword

nonn

Author

Brian Galebach, Aug 16 2000

Extensions

More terms from James A. Sellers, Aug 25 2000

Status

approved