A167928 - OEIS

%I #14 Aug 29 2016 07:29:10

%S 1,0,0,0,0,1,1,3,4,6,9,13,16,23,31,38,51,65,83,104,132,162,207,252,

%T 313,381,475,571,703,846,1032,1237,1502,1791,2164,2570,3086,3659,4375,

%U 5167,6146,7244,8584,10086,11909,13954,16421,19195,22510,26250,30696,35714

%N Number of partitions of n that do not contain 1 as a part and whose parts are not the same divisor of n.

%C Note that these partitions are located in the head of the last section of the set of partitions of n (see the shell model of partitions, here).

%H Alois P. Heinz, <a href="/A167928/b167928.txt">Table of n, a(n) for n = 0..1000</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) = A002865(n) - A032741(n).

%e The partitions of 6 are:

%e 6 ....................... All parts are the same divisor of n.

%e 5 + 1 ................... Contains 1 as a part.

%e 4 + 2 ................... All parts are not the same divisor of n. <------(1)

%e 4 + 1 + 1 ............... Contains 1 as a part.

%e 3 + 3 ................... All parts are the same divisor of n.

%e 3 + 2 + 1 ............... Contains 1 as a part.

%e 3 + 1 + 1 + 1 ........... Contains 1 as a part.

%e 2 + 2 + 2 ............... All parts are the same divisor of n.

%e 2 + 2 + 1 + 1 ........... Contains 1 as a part.

%e 2 + 1 + 1 + 1 + 1 ....... Contains 1 as a part.

%e 1 + 1 + 1 + 1 + 1 + 1 ... Contains 1 as a part.

%e Then a(6) = 1.

%p b:= proc(n, i, t) option remember;

%p       `if`(n=0, `if`(t<>1, 1, 0), `if`(i<2, 0,

%p       add(b(n-i*j, i-1, `if`(j=0, t, max(0, t-1))), j=0..n/i)))

%p     end:

%p a:= n-> b(n, n, 2):

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, May 24 2013

%t Prepend[Array[ n \[Function] Length@Select[IntegerPartitions[n, All, Range[2, n - 1]], Length[Union[ # ]] > 1 &], 40], 1] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t != 1, 1, 0], If[i < 2, 0, Sum[b[n - i*j, i - 1, If[j == 0, t, Max[0, t - 1]]], {j, 0, n/i}]]]; a[n_] := b[n, n, 2]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)

%Y Cf. A000041, A002865, A032741, A144300, A135010, A138121, A167929, A167930, A167932, A167934.

%K nonn

%O 0,8

%A _Omar E. Pol_, Nov 17 2009

%E More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

%E More terms from _Alois P. Heinz_, May 24 2013