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Number of partitions of n such that the largest part is a multiple of the smallest part.
7

%I #19 Sep 03 2017 17:20:46

%S 1,2,3,5,6,11,12,20,26,37,45,71,84,117,152,203,253,342,421,556,694,

%T 884,1096,1409,1729,2168,2672,3327,4061,5039,6114,7514,9110,11098,

%U 13400,16275,19537,23575,28245,33929,40465,48424,57552,68569,81296,96449

%N Number of partitions of n such that the largest part is a multiple of the smallest part.

%C Also number of partitions of n such that the number of parts is a multiple of the multiplicity of the largest part. Example: a(7)=12 because from the 15 (=A000041(7)) partitions of 7 only [3,3,1], [2,2,2,1] and [2,2,1,1,1] do not qualify (3,4,5 are not multiples of 2,3,2, respectively). - _Emeric Deutsch_, Apr 21 2006

%H Alois P. Heinz, <a href="/A117086/b117086.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: Sum_{L>=0} Sum_{k>=1} (x^((L+1)*k) / Product_{i=k..L*k} (1 - x^i)).

%e a(7)=12 because from the 15 (=A000041(7)) partitions of 7 only [5,2],[4,3] and [3,2,2] do not qualify.

%p f:=add(add(x^((l+1)*k)/mul(1-x^i,i=k..l*k),k=1..51),l=0..51): s:=series(f,x,51):for m from 1 to 50 do c:=coeff(s,x,m): printf(`%d,`,c);od: # (Jovovic) - _Emeric Deutsch_, Apr 21 2006

%t Table[Count[IntegerPartitions[n],_?(Divisible[First[#],Last[#]]&)], {n,50}] (* _Harvey P. Dale_, Mar 04 2012 *)

%Y Cf. A118096.

%Y Cf. A000041.

%K easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Apr 17 2006

%E More terms from _Emeric Deutsch_, Apr 21 2006