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
Examples
a(42) = #{2+3+37, 2+11+29, 2+17+23} = 3.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
Crossrefs
Programs
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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 *) -
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
Formula
From Alois P. Heinz, Nov 22 2012: (Start)
G.f.: Sum_{0
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
Comments