A167932 - OEIS

%I #11 Mar 11 2014 01:34:16

%S 1,1,2,3,4,4,7,6,9,10,13,13,20,19,25,30,36,39,51,55,69,79,92,105,129,

%T 144,168,195,227,257,303,341,395,451,515,588,676,761,867,985,1120,

%U 1261,1433,1611,1821,2053,2307,2591,2919,3266,3663,4100,4587,5121,5725,6381

%N Number of partitions of n such that all parts are equal or all parts are distinct.

%C Note that for positive integers the number of partitions of n such that all parts are equal is equal to the number of proper divisors of n. (A032741(n)).

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the shell model of partitions (2D and 3D view)</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>

%F a(n) = A000041(n) - A167930(n).

%F a(n) = A000009(n) + A032741(n).

%e The partitions of 6 are:

%e 6 .............. All parts are distinct ..... (1).

%e 5+1 ............ All parts are distinct ..... (2).

%e 4+2 ............ All parts are distinct ..... (3).

%e 4+1+1 .......... Only some parts are equal.

%e 3+3 ............ All parts are equal ........ (4).

%e 3+2+1 .......... All parts are distinct ..... (5).

%e 3+1+1+1 ........ Only some parts are equal.

%e 2+2+2 .......... All parts are equal ........ (6).

%e 2+2+1+1 ........ Only some parts are equal.

%e 2+1+1+1+1 ...... Only some parts are equal.

%e 1+1+1+1+1+1 .... All parts are equal ........ (7).

%e So a(6) = 7.

%t ds[n_]:=Module[{lun=Length[Union[n]]},Length[n]==lun||lun==1]; Table[ Count[ IntegerPartitions[n],_?(ds)],{n,0,60}] (* _Harvey P. Dale_, Sep 13 2011 *)

%Y Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167930, A167931, A167933.

%K nonn

%O 0,3

%A _Omar E. Pol_, Nov 15 2009

%E More terms from _D. S. McNeil_, May 10 2010