A045450 Number of partitions of n into a prime number of distinct prime parts.

1, 0, 1, 1, 1, 2, 0, 2, 1, 2, 2, 3, 0, 4, 2, 4, 3, 4, 2, 5, 3, 5, 3, 5, 3, 6, 5, 5, 5, 7, 5, 9, 5, 7, 8, 8, 6, 11, 8, 11, 9, 12, 10, 14, 11, 15, 12, 15, 13, 18, 17, 17, 16, 18, 18, 23, 20, 22, 23, 25, 23, 30, 26, 28, 29, 32, 32, 36, 34, 38, 38, 41, 41, 47, 45, 47, 48, 50, 54, 58, 57, 60, 63

Offset: 5

Author

Vladeta Jovovic, Jul 21 2003

Keywords

• nonn

Examples

			a(50) = 15 because there are 15 partitions of 50 into a prime number of distinct prime parts: 2+7+11+13+17 = 2+5+11+13+19 = 2+5+7+17+19 = 2+5+7+13+23 = 2+3+5+17+23 = 2+3+5+11+29 = 2+19+29 = 2+17+31 = 2+11+37 = 2+7+41 = 2+5+43 = 19+31 = 13+37 = 7+43 = 3+47.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..5000

Crossrefs

Cf. A000586.

Programs

Maple

s:= proc(n) if n<1 then 0 else ithprime(n)+s(n-1) fi end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(s(i) `if`(p>n, 0, x*b(n-p, i-1)))(ithprime(i)))))
    end:
a:= n-> (p-> add(`if`(isprime(i), coeff(p, x, i), 0)
         , i=2..degree(p)))(b(n, numtheory[pi](n))):
seq(a(n), n=5..100);  # Alois P. Heinz, Sep 18 2017

Mathematica

partprim[n_] := Module[{sp, spq, sps},
sp = Subsets[Prime[Range[PrimePi[n]]]];
spq = Select[sp, PrimeQ@Length@# &];
sps = Select[spq, n == Plus@@# &];
sps // Length // Return];
Table[partprim[n], {n, 5, 80}] (* Andres Cicuttin, Sep 17 2017 *)
s[n_] := s[n] = If [n < 1, 0, Prime[n] + s[n - 1]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[s[i] < n, 0, b[n, i - 1] + Function[p, If[p > n, 0, x*b[n - p, i - 1]]][Prime[i]]]]];
a[n_] := Function[p, Sum[If[PrimeQ[i], Coefficient[p, x, i], 0], {i, 2, Exponent[p, x]}]][b[n, PrimePi[n]]];
Table[a[n], {n, 5, 100}] (* Jean-François Alcover, Jun 11 2021, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(AllTrue[#,PrimeQ]&&Length[#]==Length[ Union[ #]] && PrimeQ[Length[#]]&)],{n,5,90}] (* _Harvey P. Dale, May 17 2024 *)