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%I #27 Feb 01 2018 10:16:41
%S 1,1,1,2,3,6,8,14,19,28,39,55,72,100,132,173,227,296,380,489,622,789,
%T 999,1254,1568,1956,2433,3007,3713,4564,5597,6841,8344,10140,12307,
%U 14880,17969,21636,26012,31182,37331,44582,53167,63260,75170
%N a(n) = A000041(n) - A032741(n).
%C a(n) is also the number of partitions of n whose parts are not all equal, (including however the partition with a single part of size n). Note that the number of partitions of n whose parts are all equal gives the number of divisors of n, for n>0. (See also A144300.)
%H Alois P. Heinz, <a href="/A167934/b167934.txt">Table of n, a(n) for n = 0..10000</a>
%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) - A032741(n).
%e The partitions of n = 6 are:
%e 6 ....................... All parts are equal, but included .. (1).
%e 5 + 1 ................... All parts are not equal ............ (2).
%e 4 + 2 ................... All parts are not equal ............ (3).
%e 4 + 1 + 1 ............... All parts are not equal ............ (4).
%e 3 + 3 ................... All parts are equal, not included.
%e 3 + 2 + 1 ............... All parts are not equal ............ (5).
%e 3 + 1 + 1 + 1 ........... All parts are not equal ............ (6).
%e 2 + 2 + 2 ............... All parts are equal, not included.
%e 2 + 2 + 1 + 1 ........... All parts are not equal ............ (7).
%e 2 + 1 + 1 + 1 + 1 ....... All parts are not equal ............ (8).
%e 1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal, not included.
%e Then a(6) = 8.
%p b:= proc(n, i, k) option remember;
%p if n<0 then 0
%p elif n=0 then `if`(k=0, 1, 0)
%p elif i=0 then 0
%p else b(n, i-1, k)+
%p b(n-i, i, `if`(k<0, i, `if`(k<>i, 0, k)))
%p fi
%p end:
%p a:= n-> 1 +b(n, n-1, -1):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Dec 01 2010
%t a[0] = 1; a[n_] := PartitionsP[n] - DivisorSigma[0, n] + 1; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jan 08 2016 *)
%Y Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A144300, A135010, A138121, A167930, A167932, A167935.
%K nonn
%O 0,4
%A _Omar E. Pol_, Nov 16 2009
%E More terms from _Alois P. Heinz_, Dec 01 2010
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