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Number of strict integer partitions of n with a part dividing all the other parts.
26

%I #23 Dec 02 2023 11:04:37

%S 1,1,2,2,2,4,3,5,5,7,6,12,9,13,15,20,18,28,26,37,39,47,49,71,68,85,94,

%T 117,120,159,160,201,216,257,277,348,357,430,470,562,592,720,758,901,

%U 981,1134,1220,1457,1542,1798,1952,2250,2419,2819,3023,3482,3773,4291

%N Number of strict integer partitions of n with a part dividing all the other parts.

%C If n > 0, we can assume such a part is the smallest. - _Gus Wiseman_, Apr 23 2021

%C 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

%F a(n) = Sum_{d|n} A025147(d-1).

%F G.f.: Sum_{k>=1} (x^k*Product_{i>=2} (1+x^(k*i))).

%e From _Gus Wiseman_, Dec 01 2023: (Start)

%e The a(1) = 1 through a(8) = 5 strict partitions with a part dividing all the other parts:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (2,1) (3,1) (4,1) (4,2) (6,1) (6,2)

%e (5,1) (4,2,1) (7,1)

%e (3,2,1) (4,3,1)

%e (5,2,1)

%e The a(1) = 1 through a(8) = 5 uniform partitions containing 1:

%e (1) (11) (21) (31) (41) (51) (61) (71)

%e (111) (1111) (11111) (321) (421) (431)

%e (2211) (1111111) (521)

%e (111111) (3311)

%e (11111111)

%e (End)

%t 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 *)

%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Or@@Table[And@@IntegerQ/@(#/x), {x,#}]&]], {n,0,30}] (* _Gus Wiseman_, Apr 23 2021 *)

%Y The non-strict version is A083710.

%Y The case with no 1's is A098965.

%Y The Heinz numbers of these partitions are A339563.

%Y The strict complement is counted by A341450.

%Y The version for "divisible by" instead of "dividing" is A343347.

%Y The case where there is also a part divisible by all the others is A343378.

%Y The case where there is no part divisible by all the others is A343381.

%Y A000005 counts divisors.

%Y A000009 counts strict partitions.

%Y A000070 counts partitions with a selected part.

%Y A006128 counts partitions with a selected position.

%Y A015723 counts strict partitions with a selected part.

%Y A018818 counts partitions into divisors (strict: A033630).

%Y A167865 counts strict chains of divisors > 1 summing to n.

%Y Cf. A083711, A098743, A130689, A200745, A264401, A338470, A343377, A343379, A343380.

%Y Cf. A023645, A025147, A047966, A072774, A096765, A329436, A367586.

%K easy,nonn

%O 1,3

%A _Vladeta Jovovic_, Oct 23 2004

%E More terms from _Robert G. Wilson v_, Nov 01 2004

%E Name shortened by _Gus Wiseman_, Apr 23 2021