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A326982
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Total sum of composite parts in all partitions of n.
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3
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0, 0, 0, 0, 4, 4, 14, 18, 44, 67, 117, 166, 283, 391, 603, 848, 1250, 1702, 2442, 3280, 4565, 6094, 8266, 10878, 14566, 18970, 24953, 32255, 41909, 53619, 68983, 87542, 111496, 140561, 177436, 222125, 278425, 346293, 430951, 533083, 659268, 810948, 997322
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OFFSET
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0,5
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LINKS
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FORMULA
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EXAMPLE
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For n = 6 we have:
--------------------------------------
Partitions Sum of
of 6 composite parts
--------------------------------------
6 .......................... 6
3 + 3 ...................... 0
4 + 2 ...................... 4
2 + 2 + 2 .................. 0
5 + 1 ...................... 0
3 + 2 + 1 .................. 0
4 + 1 + 1 .................. 4
2 + 2 + 1 + 1 .............. 0
3 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 .......... 0
1 + 1 + 1 + 1 + 1 + 1 ...... 0
--------------------------------------
Total ..................... 14
So a(6) = 14.
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0], b(n, i-1)+
(p-> p+[0, `if`(isprime(i), 0, p[1]*i)])(b(n-i, min(n-i, i))))
end:
a:= n-> b(n$2)[2]:
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MATHEMATICA
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Table[Total[Select[Flatten[IntegerPartitions[n]], CompositeQ]], {n, 0, 50}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 19 2020 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, 0}, b[n, i - 1] +
With[{p = b[n-i, Min[n-i, i]]}, p+{0, If[PrimeQ[i], 0, p[[1]]*i]}]];
a[n_] := b[n, n][[2]];
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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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