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A219107
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Number of compositions (ordered partitions) of n into distinct prime parts.
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19
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1, 0, 1, 1, 0, 3, 0, 3, 2, 2, 8, 1, 8, 3, 8, 8, 10, 25, 16, 9, 16, 38, 16, 61, 18, 62, 46, 66, 160, 91, 138, 99, 70, 122, 306, 126, 314, 151, 362, 278, 588, 901, 602, 303, 654, 1142, 888, 1759, 892, 1226, 950, 2160, 1230, 3379, 1444, 2372, 2100, 4644, 7416
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OFFSET
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0,6
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COMMENTS
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a(0) = 0 iff n in {1,4,6}.
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LINKS
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FORMULA
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EXAMPLE
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a(5) = 3: [2,3], [3,2], [5].
a(7) = 3: [2,5], [5,2], [7].
a(8) = 2: [3,5], [5,3].
a(9) = 2: [2,7], [7,2].
a(10) = 8: [2,3,5], [2,5,3], [3,2,5], [3,5,2], [5,2,3], [5,3,2], [3,7], [7,3].
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MAPLE
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with(numtheory):
b:= proc(n, i) b(n, i):=
`if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
[0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
end:
a:= proc(n) local l; l:= b(n, pi(n));
a(n):= add(l[i]*(i-1)!, i=1..nops(l))
end:
seq(a(n), n=0..70);
# second Maple program:
s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)<n, 0, `if`(n=0, t!, (p
->`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
end:
a:= n-> b(n, numtheory[pi](n), 0):
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MATHEMATICA
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zip = With[{m=Max[Length[#1], Length[#2]]}, PadRight[#1, m]+PadRight[#2, m] ]&;
b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, b[n, i-1] ~zip~ Join[{0}, If[Prime[i] > n, {}, b[n - Prime[i], i-1]]], {0}]];
a[n_] := Module[{l}, l = b[n, PrimePi[n]]; Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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