OFFSET
1,4
COMMENTS
Number of integer partitions of n with no 1's with a part dividing all the others. If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 18 2021
REFERENCES
L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{ d|n, d<n} A000041(d-1).
EXAMPLE
From Gus Wiseman, Apr 18 2021: (Start)
The a(6) = 4 through a(12) = 13 partitions:
(6) (7) (8) (9) (10) (11) (12)
(3,3) (4,4) (6,3) (5,5) (6,6)
(4,2) (6,2) (3,3,3) (8,2) (8,4)
(2,2,2) (4,2,2) (4,4,2) (9,3)
(2,2,2,2) (6,2,2) (10,2)
(4,2,2,2) (4,4,4)
(2,2,2,2,2) (6,3,3)
(6,4,2)
(8,2,2)
(3,3,3,3)
(4,4,2,2)
(6,2,2,2)
(4,2,2,2,2)
(2,2,2,2,2,2)
(End)
MAPLE
with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-1 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 2 to 100 do printf(`%d, `, a(j)) od: # James A. Sellers, Jun 21 2003
# second Maple program:
a:= n-> max(1, add(combinat[numbpart](d-1), d=numtheory[divisors](n) minus {n})):
seq(a(n), n=1..69); # Alois P. Heinz, Feb 15 2023
MATHEMATICA
a[n_] := If[n==1, 1, Sum[PartitionsP[d-1], {d, Most@Divisors[n]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 15 2023 *)
CROSSREFS
Allowing 1's gives A083710.
The strict case is A098965.
The complement (except also without 1's) is counted by A338470.
The dual version is A339619.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 16 2003
EXTENSIONS
More terms from James A. Sellers, Jun 21 2003
STATUS
approved