A303362
Number of strict integer partitions of n with pairwise indivisible parts.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 7, 7, 9, 11, 12, 13, 15, 17, 20, 23, 25, 27, 32, 35, 40, 45, 50, 55, 58, 67, 78, 84, 95, 101, 113, 124, 137, 153, 169, 180, 198, 219, 242, 268, 291, 319, 342, 374, 412, 450, 492, 535, 573, 632, 685, 746, 813, 868, 944
Offset: 1
The a(14) = 7 strict integer partitions are (14), (11,3), (10,4), (9,5), (8,6), (7,5,2), (7,4,3).
Cf.
A000009,
A000837,
A003238,
A006126,
A051424,
A259936,
A275307,
A281116,
A285572,
A285573,
A290103,
A293606,
A293993,
A303364.
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Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,60}]
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lista(nn)={local(Cache=Map());
my(excl=vector(nn, n, sumdiv(n, d, 2^(n-d))));
my(a(n, m=n, b=0)=
if(n==0, 1,
while(m>n || bittest(b,0), m--; b>>=1);
my(hk=[n, m, b], z);
if(!mapisdefined(Cache, hk, &z),
z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
mapput(Cache, hk, z)); z));
for(n=1, nn, print1(a(n), ", "))
} \\ Andrew Howroyd, Nov 02 2019
A303365
Number of integer partitions of the n-th squarefree number using squarefree numbers.
Original entry on oeis.org
1, 2, 3, 6, 9, 12, 28, 36, 60, 76, 96, 150, 228, 342, 416, 504, 877, 1484, 1759, 2079, 2885, 3387, 3968, 5413, 6304, 7328, 9852, 11395, 13159, 20082, 23056, 39532, 51385, 66488, 85660, 97078, 109907, 140465, 158573, 226918, 255268, 286920, 361606, 405470
Offset: 1
The a(5) = 9 partitions are (6), (51), (33), (321), (3111), (222), (2211), (21111), (111111).
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nn=80;
sqf=Select[Range[nn],SquareFreeQ];
ser=Product[1/(1-x^sqf[[n]]),{n,Length[sqf]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,sqf}]
A305080
Number of connected strict integer partitions of n with pairwise indivisible and squarefree parts.
Original entry on oeis.org
1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 2, 2, 3, 2, 2, 4, 2, 3, 4, 4, 3, 4, 3, 4, 5, 6, 4, 6, 5, 7, 6, 5, 6, 8, 6, 6, 6, 10, 11, 11, 9, 11, 9, 13
Offset: 1
The a(52) = 6 strict partitions together with their corresponding multiset multisystems (which are clutters):
(21,15,10,6): {{2,4},{2,3},{1,3},{1,2}}
(22,14,10,6): {{1,5},{1,4},{1,3},{1,2}}
(30,22): {{1,2,3},{1,5}}
(38,14): {{1,8},{1,4}}
(42,10): {{1,2,4},{1,3}}
(46,6): {{1,9},{1,2}}
The a(60) = 8 strict partitions together with their corresponding multiset multisystems (which are clutters):
(21,15,14,10): {{2,4},{2,3},{1,4},{1,3}}
(33,21,6): {{2,5},{2,4},{1,2}}
(35,15,10): {{3,4},{2,3},{1,3}}
(39,15,6): {{2,6},{2,3},{1,2}}
(34,26): {{1,7},{1,6}}
(38,22): {{1,8},{1,5}}
(39,21): {{2,6},{2,4}}
(46,14): {{1,9},{1,4}}
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Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,And@@SquareFreeQ/@#,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,50}]
Showing 1-3 of 3 results.
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