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A280954
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Number of integer partitions of n using predecessors of prime numbers.
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10
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1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 17, 17, 26, 26, 37, 37, 53, 53, 74, 74, 101, 101, 137, 137, 183, 183, 240, 240, 314, 314, 406, 406, 520, 520, 662, 662, 837, 837, 1049, 1049, 1311, 1311, 1627, 1627, 2008, 2008, 2469, 2469, 3021, 3021, 3678, 3678, 4466, 4466
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OFFSET
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0,3
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COMMENTS
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The predecessors of prime numbers are {1, 2, 4, 6, 10, 12, ...} = A006093.
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LINKS
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EXAMPLE
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The partitions for n=0..7 are:
(),
(1),
(2), (11),
(21),(111),
(4), (22), (211), (1111),
(41),(221),(2111),(11111),
(6), (42), (411), (222), (2211), (21111), (111111),
(61),(421),(4111),(2221),(22111),(211111),(1111111).
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i=2, 1,
b(n, prevprime(i))+`if`(i-1>n, 0, b(n-i+1, i)))
end:
a:= n-> b(n, nextprime(n)):
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(`if`(
isprime(d+1), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
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MATHEMATICA
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nn=60; invser=Series[Product[1-x^(Prime[n]-1), {n, PrimePi[nn+1]}], {x, 0, nn}];
CoefficientList[1/invser, x]
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CROSSREFS
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Even (and odd) bipartition gives A280962.
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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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