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A108572
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Number of partitions of n which, as multisets, are nontrivial repetitions of a multiset.
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4
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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, 0, 217, 0, 230, 57, 297, 20, 450, 0, 490, 102, 643, 0, 918, 0, 1004, 202, 1255, 0, 1783, 14, 1992, 298, 2438, 0, 3364, 61, 3734, 491, 4565, 0, 6251, 0, 6842, 818
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OFFSET
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1,6
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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}.
Note that A000041(25)=P(25)=1958, only 6 of which satisfy the criterion.
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MAPLE
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with(combinat):PartMulti:=proc(n::nonnegint) local count, a, i, j, b, m, k, part_vec;
bigcount:=0; if isprime(n) then return(bigcount) else ps:=partition(n); b:=nops(ps);
for m from 2 to b-1 do p:=ps[m]; a:=nops(p); part_vec:=array(1..n);
for k from 1 to n do part_vec[k]:=0 od;
for i from 1 to a do j:=p[i]; part_vec[j]:=part_vec[j]+1 od;
g:=0; for j from 1 to n do g:=igcd(g, part_vec[j]) od;
if g>1 then bigcount:=bigcount+1 fi od; return(bigcount) end if end proc;
seq(PartMulti(q), q=1..49);
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], And[Length[#]<n, GCD@@Length/@Split[#]>1]&]], {n, 20}] (* Gus Wiseman, Dec 06 2018 *)
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CROSSREFS
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Cf. A000837, A018783, A047966, A100953, A303386, A303547, A303553, A319149, A319162, A319164, A319810.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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