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A302481
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Number of partitions of n into two parts with the smaller part prime and the larger part nonprime.
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2
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0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 3, 2, 2, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 4, 3, 6, 4, 6, 3, 5, 4, 5, 3, 7, 3, 7, 6, 7, 5, 8, 4, 7, 5, 7, 5, 9, 4, 8, 5, 9, 6, 9, 4, 8, 6, 9, 6, 10, 4, 9, 8, 10, 6, 11, 5, 11, 9, 10, 6, 11, 5, 10, 7, 11, 7, 12, 5, 12, 8, 11, 8
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OFFSET
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1,11
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LINKS
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FORMULA
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a(n) = Sum_{i=1..floor(n/2)} c(i) * (1 - c(n-i)), where c is the prime characteristic (A010051).
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EXAMPLE
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a(11) = 3; 11 = 9+2 = 8+3 = 6+5, smaller parts are prime, larger nonprime.
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MATHEMATICA
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Table[Sum[(1 - (PrimePi[n - i] - PrimePi[n - i - 1])) (PrimePi[i] - PrimePi[i - 1]), {i, Floor[n/2]}], {n, 100}]
Table[Count[Boole[PrimeQ[#]&/@IntegerPartitions[n, {2}]], _?(#=={0, 1}&)], {n, 90}] (* Harvey P. Dale, Jan 05 2020 *)
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PROG
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(PARI) a(n) = sum(i=1, n\2, isprime(i)*(1-isprime(n-i))); \\ Michel Marcus, Apr 09 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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