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A192885
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A071963(n) - n, where A071963(n) is the largest prime factor of p(n), the n-th partition number A000041(n).
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4
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1, 0, 0, 0, 1, 2, 5, -2, 3, -4, -3, -4, -1, 88, -9, -4, -5, -6, -7, -12, -1, -10, 145, 228, -17, 64, 3, 16, -15, 54, 437, 280, -9, -10, 1197, 6, 17941, 244, 5, -28, 87, 152, 2375, 28, 53, 1042, 195, 20, 6965, 582, 9233, 610, 1, 5184, 5, 172, 963, 102302
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OFFSET
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0,6
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COMMENTS
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It appears that if n > 39, then a(n) is positive, i.e., A071963(n) > n. This has been checked up to n = 2500.
Cilleruelo and Luca proved that A071963(n) > log log n for almost all n, a much weaker statement. Earlier Schinzel and Wirsing proved that for all large N the product p(1)*p(2)*...*p(N) has at least C*log N distinct prime factors, for any positive constant C < 1/log 2.
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LINKS
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FORMULA
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EXAMPLE
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There are 77 partitions of 12, and 77 = 7*11, so a(12) = 11 - 12 = -1.
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MATHEMATICA
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Table[First[Last[FactorInteger[PartitionsP[n]]]] - n, {n, 0, 100}]
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PROG
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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