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

0,3

Comments

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

Links

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

Example

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).

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
# Alternative:
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]

Crossrefs

Cf. A006093, A023506.

Even (and odd) bipartition gives A280962.

Sequence in context: A341972 A277133 A323539 * A339244 A197122 A064410

Adjacent sequences: A280951 A280952 A280953 * A280955 A280956 A280957

Keyword

nonn

Author

Gus Wiseman, Jan 11 2017

Status

approved