OFFSET
1,3
COMMENTS
If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 23 2021
Also the number of uniform (constant multiplicity) partitions of n containing 1, ranked by A367586. The strict case is A096765. The version without 1 is A329436. - Gus Wiseman, Dec 01 2023
FORMULA
a(n) = Sum_{d|n} A025147(d-1).
G.f.: Sum_{k>=1} (x^k*Product_{i>=2} (1+x^(k*i))).
EXAMPLE
From Gus Wiseman, Dec 01 2023: (Start)
The a(1) = 1 through a(8) = 5 strict partitions with a part dividing all the other parts:
(1) (2) (3) (4) (5) (6) (7) (8)
(2,1) (3,1) (4,1) (4,2) (6,1) (6,2)
(5,1) (4,2,1) (7,1)
(3,2,1) (4,3,1)
(5,2,1)
The a(1) = 1 through a(8) = 5 uniform partitions containing 1:
(1) (11) (21) (31) (41) (51) (61) (71)
(111) (1111) (11111) (321) (421) (431)
(2211) (1111111) (521)
(111111) (3311)
(11111111)
(End)
MATHEMATICA
Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 62}], {k, 62}]], x], {2, 60}] (* Robert G. Wilson v, Nov 01 2004 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Or@@Table[And@@IntegerQ/@(#/x), {x, #}]&]], {n, 0, 30}] (* Gus Wiseman, Apr 23 2021 *)
CROSSREFS
The non-strict version is A083710.
The case with no 1's is A098965.
The Heinz numbers of these partitions are A339563.
The strict complement is counted by A341450.
The version for "divisible by" instead of "dividing" is A343347.
The case where there is also a part divisible by all the others is A343378.
The case where there is no part divisible by all the others is A343381.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Oct 23 2004
EXTENSIONS
More terms from Robert G. Wilson v, Nov 01 2004
Name shortened by Gus Wiseman, Apr 23 2021
STATUS
approved