cp's OEIS Frontend

A361391 Number of strict integer partitions of n with non-integer mean.

%I A361391 #11 Mar 16 2023 12:23:44
%S A361391 1,0,0,1,0,2,0,4,2,4,5,11,0,17,15,13,15,37,18,53,24,48,78,103,23,111,
%T A361391 152,143,123,255,110,339,238,372,495,377,243,759,845,873,414,1259,842,
%U A361391 1609,1383,1225,2281,2589,1285,2827,2518,3904,3836,5119,3715,4630
%N A361391 Number of strict integer partitions of n with non-integer mean.
%C A361391 Are 1, 2, 4, 6, 12 the only zeros?
%e A361391 The a(3) = 1 through a(11) = 11 partitions:
%e A361391   {2,1}  .  {3,2}  .  {4,3}    {4,3,1}  {5,4}  {5,3,2}    {6,5}
%e A361391             {4,1}     {5,2}    {5,2,1}  {6,3}  {5,4,1}    {7,4}
%e A361391                       {6,1}             {7,2}  {6,3,1}    {8,3}
%e A361391                       {4,2,1}           {8,1}  {7,2,1}    {9,2}
%e A361391                                                {4,3,2,1}  {10,1}
%e A361391                                                           {5,4,2}
%e A361391                                                           {6,3,2}
%e A361391                                                           {6,4,1}
%e A361391                                                           {7,3,1}
%e A361391                                                           {8,2,1}
%e A361391                                                           {5,3,2,1}
%p A361391 a:= proc(m) option remember; local b; b:=
%p A361391       proc(n, i, t) option remember; `if`(i*(i+1)/2<n,
%p A361391         0, `if`(n=0, signum(irem(m, t)),
%p A361391         b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
%p A361391       end: `if`(m=0, 1, b(m$2, 0))
%p A361391     end:
%p A361391 seq(a(n), n=0..55);  # _Alois P. Heinz_, Mar 16 2023
%t A361391 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!IntegerQ[Mean[#]]&]],{n,0,30}]
%Y A361391 The strict complement is counted by A102627.
%Y A361391 The non-strict version is ranked by A348551, complement A316413.
%Y A361391 The non-strict version is counted by A349156, complement A067538.
%Y A361391 For median instead of mean we have A360952, complement A359907.
%Y A361391 A000041 counts partitions, strict A000009.
%Y A361391 A008284/A058398/A327482 count partitions by mean.
%Y A361391 A307683 counts partitions with non-integer median, ranks A359912.
%Y A361391 A325347 counts partitions with integer median, ranks A359908.
%Y A361391 A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
%Y A361391 A327472 counts partitions not containing their mean, complement of A237984.
%Y A361391 A327475 counts subsets with integer mean.
%Y A361391 Cf. A051293, A082550, A143773, A175397, A175761, A240219, A240850, A326027, A326641, A326849, A359897.
%K A361391 nonn
%O A361391 0,6
%A A361391 _Gus Wiseman_, Mar 11 2023
%E A361391 a(31)-a(55) from _Alois P. Heinz_, Mar 16 2023