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A303364 Number of strict integer partitions of n with pairwise indivisible and squarefree parts.

%I A303364 #16 Dec 29 2020 03:19:48
%S A303364 1,1,1,0,2,1,2,1,1,3,2,2,4,3,3,4,6,5,5,6,7,8,9,10,10,11,11,14,14,17,
%T A303364 16,18,19,23,24,27,29,30,33,36,41,41,42,46,51,56,60,66,67,71,81,86,93,
%U A303364 96,101,110,121,129,135,144,153,159,173,192,204,207,224
%N A303364 Number of strict integer partitions of n with pairwise indivisible and squarefree parts.
%H A303364 Fausto A. C. Cariboni, <a href="/A303364/b303364.txt">Table of n, a(n) for n = 1..700</a> (terms 0..400 from Andrew Howroyd)
%e A303364 The a(23) = 9 strict integer partitions are (23), (13,10), (17,6), (21,2), (10,7,6), (11,7,5), (13,7,3), (11,7,3,2), (13,5,3,2).
%t A303364 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@SquareFreeQ/@#&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,60}]
%o A303364 (PARI)
%o A303364 lista(nn)={local(Cache=Map());
%o A303364   my(excl=vector(nn, n, sumdiv(n, d, 2^(n-d))));
%o A303364   my(c(n, m, b)=
%o A303364      if(n==0, 1,
%o A303364         while(m>n || bittest(b,0), m--; b>>=1);
%o A303364         my(hk=[n, m, b], z);
%o A303364         if(!mapisdefined(Cache, hk, &z),
%o A303364           z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
%o A303364           mapput(Cache, hk, z)); z));
%o A303364   my(a(n)=c(n, n, sum(i=1, n, if(!issquarefree(i), 2^(n-i)))));
%o A303364   for(n=1, nn, print1(a(n), ", "))
%o A303364 } \\ _Andrew Howroyd_, Nov 02 2019
%Y A303364 Cf. A000009, A000837, A003238, A005117, A006126, A051424, A073576, A285572, A285573, A293606, A293993, A303362, A303365.
%K A303364 nonn
%O A303364 1,5
%A A303364 _Gus Wiseman_, Apr 22 2018