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A125688
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Number of partitions of n into three distinct primes.
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16
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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
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OFFSET
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1,18
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COMMENTS
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LINKS
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FORMULA
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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)
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EXAMPLE
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a(42) = #{2+3+37, 2+11+29, 2+17+23} = 3.
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MAPLE
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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]:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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