cp's OEIS Frontend

A335448 Numbers whose prime indices are inseparable.

%I A335448 #11 Apr 10 2021 08:08:48
%S A335448 4,8,9,16,24,25,27,32,40,48,49,54,56,64,80,81,88,96,104,112,121,125,
%T A335448 128,135,136,144,152,160,162,169,176,184,189,192,208,224,232,240,243,
%U A335448 248,250,256,272,288,289,296,297,304,320,324,328,336,343,344,351,352
%N A335448 Numbers whose prime indices are inseparable.
%C A335448 First differs from A212164 in lacking 72.
%C A335448 First differs from A293243 in lacking 72.
%C A335448 No terms are squarefree.
%C A335448 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 A335448 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 A335448 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 A335448 A multiset is inseparable iff its maximal multiplicity is greater than one plus the sum of its remaining multiplicities.
%e A335448 The sequence of terms together with their prime indices begins:
%e A335448    4: {1,1}
%e A335448    8: {1,1,1}
%e A335448    9: {2,2}
%e A335448   16: {1,1,1,1}
%e A335448   24: {1,1,1,2}
%e A335448   25: {3,3}
%e A335448   27: {2,2,2}
%e A335448   32: {1,1,1,1,1}
%e A335448   40: {1,1,1,3}
%e A335448   48: {1,1,1,1,2}
%e A335448   49: {4,4}
%e A335448   54: {1,2,2,2}
%e A335448   56: {1,1,1,4}
%e A335448   64: {1,1,1,1,1,1}
%e A335448   80: {1,1,1,1,3}
%e A335448   81: {2,2,2,2}
%e A335448   88: {1,1,1,5}
%e A335448   96: {1,1,1,1,1,2}
%t A335448 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A335448 Select[Range[100],Select[Permutations[primeMS[#]],!MatchQ[#,{___,x_,x_,___}]&]=={}&]
%Y A335448 Complement of A335433.
%Y A335448 Separations are counted by A003242 and A335452 and ranked by A333489.
%Y A335448 Permutations of prime indices are counted by A008480.
%Y A335448 Inseparable partitions are counted by A325535.
%Y A335448 Strict permutations of prime indices are counted by A335489.
%Y A335448 Cf. A000670, A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451.
%K A335448 nonn
%O A335448 1,1
%A A335448 _Gus Wiseman_, Jun 21 2020