A168470 -id:A168470 - cp's OEIS frontend

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

Brian Galebach, Aug 16 2000

nonn

			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.

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

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

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

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

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 *)

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

More terms from James Sellers, Aug 25 2000

A153979 Prime sums of prime factors of composite(k)=A002808(k).

Original entry on oeis.org

5, 7, 7, 13, 11, 19, 11, 11, 11, 17, 11, 13, 31, 13, 13, 23, 13, 43, 17, 13, 13, 17, 19, 13, 19, 61, 23, 73, 17, 41, 23, 19, 47, 17, 19, 29, 19, 103, 29, 17, 109, 17, 19, 37, 17, 17, 71, 23, 139, 37, 19, 43, 151, 17, 83, 17, 23, 47, 43, 31, 19, 181, 17, 31, 47, 53, 193, 17, 23, 101, 23, 199, 29, 17

Offset: 1

Juri-Stepan Gerasimov, Jan 04 2009

nonn

More precisely: Take the sum of prime factors of the n-th composite number A002808(n), with repetition (e.g., 72 = 2^3*3^2 => 2+2+2+3+3). If the sum is prime, list it here; if not, don't list it and skip over to the next composite number. - M. F. Hasler, May 02 2015

The count of the same numbers is A168470. - Gionata Neri, Apr 26 2015

			A002808(1)=4=2*2, and 2+2=4(nonprime), so 4 does not contribute to this sequence. A002808(2)=6=2*3 and 2+3=5(prime), so a(1)=5. A002808(5)=10=2*5 and 2+5=7(prime), so a(2)=7. A002808(6)=12=2*2*3 and 2+2+3=7(prime), so a(3)=7.

Karl Hovekamp, Table of n, a(n) for n=1,...,12285.
Karl Hovekamp, Table of n, a(n), source, factors for n=1,...,12285.

Cf. A000040, A002808.

N:= 1000: # to get a(1) to a(N)
count:= 0:
for x from 2 while count < N do
   if not isprime(x) then
     y:= add(f[1]*f[2],f=ifactors(x)[2]);
     if isprime(y) then
       count:= count+1;
       A[count]:= y;
     fi
   fi
od;
seq(A[i],i=1..N); # Robert Israel, Apr 26 2015

lim = 410; s = Select[Range@ lim, CompositeQ]; f[n_] := Plus @@ (Flatten[Table[#1, {#2}] & @@@ FactorInteger@ n]); Select[f /@ s, PrimeQ] (* Michael De Vlieger, Apr 26 2015 *)

forcomposite(c=1,999,isprime(s=(s=factor(c))[,1]~*s[,2])&&print1(s",")) \\ M. F. Hasler, May 02 2015

Corrected and edited by Karl Hovekamp, Dec 05 2009

cp's OEIS Frontend

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

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