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A280954 Number of integer partitions of n using predecessors of prime numbers.

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