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A282467
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Number of partitions of n which are not the partitions into (one or more) consecutive parts.
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1
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0, 1, 1, 4, 5, 9, 13, 21, 27, 40, 54, 75, 99, 133, 172, 230, 295, 382, 488, 625, 788, 1000, 1253, 1573, 1955, 2434, 3006, 3716, 4563, 5600, 6840, 8348, 10139, 12308, 14879, 17974, 21635, 26013, 31181, 37336, 44581, 53170, 63259, 75173, 89128, 105556, 124752, 147271, 173522, 204223, 239939, 281587
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OFFSET
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1,4
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COMMENTS
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Also number of partitions of n minus the number of odd divisors of n.
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LINKS
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FORMULA
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EXAMPLE
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For n = 6, the number of partitions of 6 is A000041(6) = 11. There are two partitions of 6 into (one or more) consecutive parts: [6] and [3, 2, 1], so a(6) = 11 - 2 = 9. On the other hand, 6 has two odd divisors: 1 and 3, so a(6) = 11 - 2 = 9.
For n = 15, the number of partitions of 15 is A000041(15) = 176. There are four partitions of 15 into (one or more) consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1], so a(15) = 176 - 4 = 172. On the other hand, 15 has four odd divisors: 1, 3, 5 and 15, so a(15) = 176 - 4 = 172.
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MAPLE
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a:= n-> combinat[numbpart](n)-mul(`if`(i[1]=2, 1, i[2]+1), i=ifactors(n)[2]):
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MATHEMATICA
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Table[PartitionsP@ n - DivisorSum[n, Boole[# > 0] &, OddQ@ # &], {n, 52}] (* Michael De Vlieger, Feb 27 2017 *)
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PROG
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(Sage)
A282467 = lambda n: number_of_partitions(n) - len(list(filter(is_odd, divisors(n))))
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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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