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A305563
Number of reducible integer partitions of n.
29
1, 2, 3, 4, 7, 7, 15, 16, 27, 30, 56, 56, 100, 105, 157, 188, 287, 303, 470, 524, 724, 850, 1197, 1339, 1856, 2135, 2814, 3305, 4360, 4951, 6532, 7561, 9563, 11195, 14165, 16328, 20631, 23866, 29471, 34320, 42336, 48672, 59872, 69139, 83625, 96911, 117153
OFFSET
1,2
COMMENTS
A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is reducible if either m is of size 1 or gcd(m_1, ..., m_k) = 1 and the multiset {y_1, ..., y_k} is also reducible.
EXAMPLE
The a(6) = 7 reducible integer partitions are (6), (51), (411), (321), (3111), (21111), (111111). Missing from this list are (42), (33), (222), (2211).
MATHEMATICA
ptnredQ[y_]:=Or[Length[y]==1, And[GCD@@y==1, ptnredQ[Sort[Length/@Split[y], Greater]]]];
Table[Length[Select[IntegerPartitions[n], ptnredQ]], {n, 20}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 05 2018
STATUS
approved