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A056768
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Number of partitions of the n-th prime into parts that are all primes.
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23
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1, 1, 2, 3, 6, 9, 17, 23, 40, 87, 111, 219, 336, 413, 614, 1083, 1850, 2198, 3630, 5007, 5861, 9282, 12488, 19232, 33439, 43709, 49871, 64671, 73506, 94625, 221265, 279516, 394170, 441250, 766262, 853692, 1175344, 1608014, 1975108, 2675925
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = [x^prime(n)] Product_{k>=1} 1/(1 - x^prime(k)). - Ilya Gutkovskiy, Jun 05 2017
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EXAMPLE
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a(4) = 3 because the 4th prime is 7 which can be partitioned using primes in 3 ways: 7, 5 + 2, and 3 + 2 + 2.
In connection with the 6th prime 13, for instance, we have the a(6) = 9 prime partitions: 13 = 2 + 2 + 2 + 2 + 2 + 3 = 2 + 2 + 2 + 2 + 5 = 2 + 2 + 2 + 7 = 2 + 2 + 3 + 3 + 3 = 2 + 3 + 3 + 5 = 2 + 11 = 3 + 3 + 7 = 3 + 5 + 5.
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i=2
and n::even, 1, `if`(i=2 or n=1, 0,
b(n, prevprime(i)))+`if`(i>n, 0, b(n-i, i)))
end:
a:= n-> b(ithprime(n)$2):
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MATHEMATICA
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Table[Count[IntegerPartitions[n], _?(AllTrue[#, PrimeQ]&)], {n, Prime[ Range[ 40]]}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 07 2015 *)
n=40; ser=Product[1/(1-x^Prime[i]), {i, 1, n}]; Table[SeriesCoefficient[ser, {x, 0, Prime[i]}], {i, 1, n}] (* Gus Wiseman, Sep 14 2016 *)
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PROG
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(Haskell)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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