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

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

Offset: 0

Author

Gus Wiseman, Jan 11 2017

Keywords

• nonn

Comments

The predecessors of prime numbers are {1, 2, 4, 6, 10, 12, ...} = A006093.

Examples

			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).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A006093, A023506.

Even (and odd) bipartition gives A280962.

Programs

Maple

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)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 11 2017
# 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:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

Mathematica

nn=60;invser=Series[Product[1-x^(Prime[n]-1),{n,PrimePi[nn+1]}],{x,0,nn}];
CoefficientList[1/invser,x]