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

5,6

Links

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

Example

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.

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)<n, 0,
       b(n, i-1)+(p-> `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 *)

Crossrefs

Cf. A000586.

Sequence in context: A249072 A174007 A330709 * A347419 A029222 A162350

Adjacent sequences: A045447 A045448 A045449 * A045451 A045452 A045453

Keyword

nonn

Author

Vladeta Jovovic, Jul 21 2003

Status

approved