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%I #12 Jun 07 2018 11:00:15
%S 1,1,2,2,4,4,7,7,11,11,17,17,26,26,37,37,53,53,74,74,101,101,137,137,
%T 183,183,240,240,314,314,406,406,520,520,662,662,837,837,1049,1049,
%U 1311,1311,1627,1627,2008,2008,2469,2469,3021,3021,3678,3678,4466,4466
%N Number of integer partitions of n using predecessors of prime numbers.
%C The predecessors of prime numbers are {1, 2, 4, 6, 10, 12, ...} = A006093.
%H Alois P. Heinz, <a href="/A280954/b280954.txt">Table of n, a(n) for n = 0..10000</a>
%e The partitions for n=0..7 are:
%e (),
%e (1),
%e (2), (11),
%e (21),(111),
%e (4), (22), (211), (1111),
%e (41),(221),(2111),(11111),
%e (6), (42), (411), (222), (2211), (21111), (111111),
%e (61),(421),(4111),(2221),(22111),(211111),(1111111).
%p b:= proc(n, i) option remember; `if`(n=0 or i=2, 1,
%p b(n, prevprime(i))+`if`(i-1>n, 0, b(n-i+1, i)))
%p end:
%p a:= n-> b(n, nextprime(n)):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jan 11 2017
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(`if`(
%p isprime(d+1), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p end:
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jun 07 2018
%t nn=60;invser=Series[Product[1-x^(Prime[n]-1),{n,PrimePi[nn+1]}],{x,0,nn}];
%t CoefficientList[1/invser,x]
%Y Cf. A006093, A023506.
%Y Even (and odd) bipartition gives A280962.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jan 11 2017