A280954 - OEIS

%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