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%I #9 Apr 22 2021 22:01:31
%S 1,2,3,5,6,10,11,16,19,26,27,41,42,55,64,81,83,114,116,151,168,202,
%T 210,277,289,348,382,460,478,604,623,747,812,942,1006,1223,1269,1479,
%U 1605,1870,1959,2329,2434,2818,3056,3458,3653,4280,4493,5130,5507,6231,6580
%N Number of partitions of n such that every part divides the largest part and such that the smallest part divides every part.
%C First differs from A130689 at a(11) = 27, A130689(11) = 28.
%C Alternative name: Number of integer partitions of n with a part divisible by and a part dividing all the other parts. With this definition we have a(0) = 1. - _Gus Wiseman_, Apr 18 2021
%F G.f.: Sum_{i>=0} Sum_(j>0} x^(j+i*j)/Product_{k|i} (1-x^(j*k)).
%e From _Gus Wiseman_, Apr 18 2021: (Start)
%e The a(1) = 1 though a(8) = 16 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (21) (22) (41) (33) (61) (44)
%e (111) (31) (221) (42) (331) (62)
%e (211) (311) (51) (421) (71)
%e (1111) (2111) (222) (511) (422)
%e (11111) (411) (2221) (611)
%e (2211) (4111) (2222)
%e (3111) (22111) (3311)
%e (21111) (31111) (4211)
%e (111111) (211111) (5111)
%e (1111111) (22211)
%e (41111)
%e (221111)
%e (311111)
%e (2111111)
%e (11111111)
%e (End)
%p A130714 := proc(n) local gf,den,i,k,j ; gf := 0 ; for i from 0 to n do for j from 1 to n/(1+i) do den := 1 ; for k in numtheory[divisors](i) do den := den*(1-x^(j*k)) ; od ; gf := taylor(gf+x^(j+i*j)/den,x=0,n+1) ; od ; od: coeftayl(gf,x=0,n) ; end: seq(A130714(n),n=1..60) ; # _R. J. Mathar_, Oct 28 2007
%t Table[If[n==0,1,Length[Select[IntegerPartitions[n],And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]]],{n,0,30}] (* _Gus Wiseman_, Apr 18 2021 *)
%Y The second condition alone gives A083710.
%Y The first condition alone gives A130689.
%Y The opposite version is A343342.
%Y The Heinz numbers of these partitions are the complement of A343343.
%Y The half-opposite versions are A343344 and A343345.
%Y The complement is counted by A343346.
%Y The strict case is A343378.
%Y A000009 counts strict partitions.
%Y A000041 counts 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 Cf. A338470, A341450, A342193, A343337, A343338, A343341, A343379, A343382.
%K easy,nonn
%O 1,2
%A _Vladeta Jovovic_, Jul 02 2007
%E More terms from _R. J. Mathar_, Oct 28 2007