cp's OEIS Frontend

A386632 Numbers k such that there is a disjoint inseparable way to choose a strict integer partition of each exponent in the prime factorization of k.

%I A386632 #13 Aug 13 2025 22:25:35
%S A386632 8,16,27,32,64,81,125,128,243,256,343,512,625,729,1024,1331,1536,2048,
%T A386632 2187,2197,2304,2401,2560,3072,3125,3456,3584,4096,4608,4913,5120,
%U A386632 5184,5632,6144,6400,6561,6656,6859,6912,7168,8192,8704,9216,9728,10240,11264
%N A386632 Numbers k such that there is a disjoint inseparable way to choose a strict integer partition of each exponent in the prime factorization of k.
%C A386632 First cubefull number (A246549) not in this sequence is 216.
%C A386632 The first term that is not a prime power is 1536.
%C A386632 A set partition is inseparable iff the underlying set has no permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
%e A386632 The prime indices of 2304 are {1,1,1,1,1,1,1,1,2,2}, and we have disjoint inseparable choice {{4,3,1},{2}}, so 2304 is in the sequence.
%e A386632 The terms together with their prime indices begin:
%e A386632      8: {1,1,1}
%e A386632     16: {1,1,1,1}
%e A386632     27: {2,2,2}
%e A386632     32: {1,1,1,1,1}
%e A386632     64: {1,1,1,1,1,1}
%e A386632     81: {2,2,2,2}
%e A386632    125: {3,3,3}
%e A386632    128: {1,1,1,1,1,1,1}
%e A386632    243: {2,2,2,2,2}
%e A386632    256: {1,1,1,1,1,1,1,1}
%e A386632    343: {4,4,4}
%e A386632    512: {1,1,1,1,1,1,1,1,1}
%e A386632    625: {3,3,3,3}
%e A386632    729: {2,2,2,2,2,2}
%t A386632 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A386632 dsj[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
%t A386632 insepQ[y_]:=2*Max[y]>Total[y]+1;
%t A386632 Join@@Position[Sign[Table[Length[Select[dsj[prix[n]],insepQ[Length/@#]&]],{n,1000}]],1]
%Y A386632 This is the inseparable case of A351294, positives in A386575, counted by A239455.
%Y A386632 Also positions of positive terms in A386582.
%Y A386632 A000110 counts set partitions, ordered A000670.
%Y A386632 A003242 and A335452 count separations, ranks A333489.
%Y A386632 A025065/A386638 counts inseparable type partitions, ranks A335126, sums of A386586.
%Y A386632 A325534 counts separable multisets, ranks A335433, sums of A386583.
%Y A386632 A325535 counts inseparable multisets, ranks A335448, sums of A386584.
%Y A386632 A336106 counts separable type partitions, ranks A335127, sums of A386585.
%Y A386632 A386633 counts separable type set partitions, row sums of A386635.
%Y A386632 A386634 counts inseparable type set partitions, row sums of A386636.
%Y A386632 Cf. A001221, A001222, A051903, A051904, A056239, A130091, A279790, A351293, A373957, A382525, A386587.
%K A386632 nonn
%O A386632 1,1
%A A386632 _Gus Wiseman_, Aug 04 2025