A108572 - OEIS

%I #9 Dec 07 2018 07:54:43

%S 0,0,0,1,0,3,0,4,2,7,0,13,0,15,8,21,0,37,0,44,16,56,0,93,6,101,29,137,

%T 0,217,0,230,57,297,20,450,0,490,102,643,0,918,0,1004,202,1255,0,1783,

%U 14,1992,298,2438,0,3364,61,3734,491,4565,0,6251,0,6842,818

%N Number of partitions of n which, as multisets, are nontrivial repetitions of a multiset.

%C The singleton and the all-ones partitions are ignored, so that a(n)=0 if n is prime. If a partition is listed as m_1^am_2^bm_3^c..., then it is counted exactly when gcd(a,b,c,...)>1. These are equinumerous (conjugate) with those partitions for which gcd(m_1,m_2,...)>1 (less 1, the singleton), hence the formula.

%F a(n) = A018783(n)-1, n>1. - _Vladeta Jovovic_, Jul 28 2005

%e a(25) = 6: 1^(15)2^5 = 5{1, 1, 1, 2}, 1^52^(10) = 5{1, 2, 2}, 1^(10)3^5 = 5{3, 1, 1}, 2^53^5 = 5{3, 2}, 1^44^4 = 5{4, 1}, 5^5 = 5{5}.

%e Note that A000041(25)=P(25)=1958, only 6 of which satisfy the criterion.

%p with(combinat):PartMulti:=proc(n::nonnegint) local count,a,i,j,b,m,k,part_vec;

%p bigcount:=0; if isprime(n) then return(bigcount) else ps:=partition(n); b:=nops(ps);

%p for m from 2 to b-1 do p:=ps[m]; a:=nops(p); part_vec:=array(1..n);

%p for k from 1 to n do part_vec[k]:=0 od;

%p for i from 1 to a do j:=p[i]; part_vec[j]:=part_vec[j]+1 od;

%p g:=0; for j from 1 to n do g:=igcd(g,part_vec[j]) od;

%p if g>1 then bigcount:=bigcount+1 fi od; return(bigcount) end if end proc;

%p seq(PartMulti(q),q=1..49);

%t Table[Length[Select[IntegerPartitions[n],And[Length[#]<n,GCD@@Length/@Split[#]>1]&]],{n,20}] (* _Gus Wiseman_, Dec 06 2018 *)

%Y Cf. A000837, A018783, A047966, A100953, A303386, A303547, A303553, A319149, A319162, A319164, A319810.

%K nonn

%O 1,6

%A _Len Smiley_, Jul 25 2005

%E More terms from _Gus Wiseman_, Dec 06 2018