cp's OEIS Frontend

A323867 Number of aperiodic arrays of positive integers summing to n.

%I A323867 #9 Aug 22 2019 23:11:14
%S A323867 1,1,1,5,11,33,57,157,303,683,1358,2974,5932,12560,25328,52400,106256,
%T A323867 217875,441278,899955,1822703,3701401,7491173,15178253,30691135,
%U A323867 62085846,125435689,253414326,511547323,1032427635,2082551931,4199956099,8466869525,17064777665
%N A323867 Number of aperiodic arrays of positive integers summing to n.
%C A323867 The 1-dimensional case is A000740.
%C A323867 An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.
%H A323867 Andrew Howroyd, <a href="/A323867/b323867.txt">Table of n, a(n) for n = 0..200</a>
%e A323867 The a(5) = 33 arrays:
%e A323867   5  14  23  32  41  113  122  131  212  221  311  1112  1121  1211  2111
%e A323867 .
%e A323867   1  2  3  4  11  11  12  21
%e A323867   4  3  2  1  12  21  11  11
%e A323867 .
%e A323867   1  1  1  2  2  3
%e A323867   1  2  3  1  2  1
%e A323867   3  2  1  2  1  1
%e A323867 .
%e A323867   1  1  1  2
%e A323867   1  1  2  1
%e A323867   1  2  1  1
%e A323867   2  1  1  1
%t A323867 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323867 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323867 ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
%t A323867 apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
%t A323867 Table[Length[Union@@Table[Select[ptnmats[k],apermatQ],{k,Times@@Prime/@#&/@IntegerPartitions[n]}]],{n,15}]
%o A323867 (GAP) List([0..30], A323867); # See A323861 for code; _Andrew Howroyd_, Aug 21 2019
%Y A323867 Cf. A000670, A000740, A027375, A101509.
%Y A323867 Cf. A323860, A323861, A323862, A323863, A323864, A323866, A323869.
%K A323867 nonn
%O A323867 0,4
%A A323867 _Gus Wiseman_, Feb 04 2019
%E A323867 Terms a(16) and beyond from _Andrew Howroyd_, Aug 21 2019