|
%I #11 Apr 10 2021 08:08:48
%S 4,8,9,16,24,25,27,32,40,48,49,54,56,64,80,81,88,96,104,112,121,125,
%T 128,135,136,144,152,160,162,169,176,184,189,192,208,224,232,240,243,
%U 248,250,256,272,288,289,296,297,304,320,324,328,336,343,344,351,352
%N Numbers whose prime indices are inseparable.
%C First differs from A212164 in lacking 72.
%C First differs from A293243 in lacking 72.
%C No terms are squarefree.
%C Also Heinz numbers of inseparable partitions (A325535). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C These are also numbers that can be written as a product of prime numbers, each different from the last but not necessarily different from those prior to the last.
%C A multiset is inseparable iff its maximal multiplicity is greater than one plus the sum of its remaining multiplicities.
%e The sequence of terms together with their prime indices begins:
%e 4: {1,1}
%e 8: {1,1,1}
%e 9: {2,2}
%e 16: {1,1,1,1}
%e 24: {1,1,1,2}
%e 25: {3,3}
%e 27: {2,2,2}
%e 32: {1,1,1,1,1}
%e 40: {1,1,1,3}
%e 48: {1,1,1,1,2}
%e 49: {4,4}
%e 54: {1,2,2,2}
%e 56: {1,1,1,4}
%e 64: {1,1,1,1,1,1}
%e 80: {1,1,1,1,3}
%e 81: {2,2,2,2}
%e 88: {1,1,1,5}
%e 96: {1,1,1,1,1,2}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],Select[Permutations[primeMS[#]],!MatchQ[#,{___,x_,x_,___}]&]=={}&]
%Y Complement of A335433.
%Y Separations are counted by A003242 and A335452 and ranked by A333489.
%Y Permutations of prime indices are counted by A008480.
%Y Inseparable partitions are counted by A325535.
%Y Strict permutations of prime indices are counted by A335489.
%Y Cf. A000670, A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 21 2020
|
