A125688 Number of partitions of n into three distinct primes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 4, 3, 4, 2, 5, 3, 5, 4, 6, 1, 6, 3, 6, 4, 6, 3, 9, 3, 8, 5, 8, 4, 11, 3, 11, 5, 10, 3, 13, 3, 13, 6, 12, 2, 14, 5, 15, 6, 13, 2, 18, 5, 17, 6, 14, 4, 21, 5, 19, 7, 17, 4, 25, 4, 20, 8, 21, 4, 26, 4, 25, 9, 22, 4

Offset

1,18

Comments

a(A124868(n)) = 0; a(A124867(n)) > 0;

a(A125689(n)) = n and a(m) <> n for m < A125689(n).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

Formula

From Alois P. Heinz, Nov 22 2012: (Start)

G.f.: Sum_{0<i_1<i_2<i_3} x^(Sum_{j=1..3} prime(i_j)).

a(n) = [x^n*y^3] Product_{i>=1} (1+x^prime(i)*y). (End)

a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} A010051(i) * A010051(k) * A010051(n-i-k). - Wesley Ivan Hurt, Mar 29 2019

Example

a(42) = #{2+3+37, 2+11+29, 2+17+23} = 3.

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3], `if`(i<1, [0$4],
       zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$3],
       b(n-ithprime(i), i-1)[1..3])[]], 0)))
    end:
a:= n-> b(n, numtheory[pi](n))[4]:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 15 2012

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i<1, {0, 0, 0, 0}, Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, {0, 0, 0}, Take[b[n-Prime[i], i-1], 3]]]}]]]; a[n_] := b[n, PrimePi[n]][[4]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)
dp3Q[{a_, b_, c_}]:=Length[Union[{a, b, c}]]==3&&AllTrue[{a, b, c}, PrimeQ]; Table[ Count[IntegerPartitions[n, {3}], _?dp3Q], {n, 100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 30 2019 *)

Prog

(PARI) a(n)=my(s); forprime(p=n\3, n-4, forprime(q=(n-p)\2+1, min(n-p, p-1), if(isprime(n-p-q), s++))); s \\ Charles R Greathouse IV, Aug 27 2012

Crossrefs

Column k=3 of A219180. - Alois P. Heinz, Nov 13 2012

Cf. A010051, A068307, A124868, A125689.

Sequence in context: A339221 A025851 A343911 * A230257 A060508 A029404

Adjacent sequences: A125685 A125686 A125687 * A125689 A125690 A125691

Keyword

nonn,look

Author

Reinhard Zumkeller, Nov 30 2006

Status

approved