cp's OEIS Frontend

A335433 Numbers whose multiset of prime indices is separable.

%I A335433 #5 Jul 03 2020 06:58:44
%S A335433 1,2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,26,28,29,30,31,33,
%T A335433 34,35,36,37,38,39,41,42,43,44,45,46,47,50,51,52,53,55,57,58,59,60,61,
%U A335433 62,63,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,82,83
%N A335433 Numbers whose multiset of prime indices is separable.
%C A335433 First differs from A212167 in having 72.
%C A335433 Includes all squarefree numbers A005117.
%C A335433 A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
%C A335433 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 A335433 Also Heinz numbers of separable partitions (A325534). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A335433 Also numbers that cannot be written as a product of prime numbers, each different from the last but not necessarily different from those prior to the last.
%e A335433 The sequence of terms together with their prime indices begins:
%e A335433       1: {}          20: {1,1,3}       39: {2,6}
%e A335433       2: {1}         21: {2,4}         41: {13}
%e A335433       3: {2}         22: {1,5}         42: {1,2,4}
%e A335433       5: {3}         23: {9}           43: {14}
%e A335433       6: {1,2}       26: {1,6}         44: {1,1,5}
%e A335433       7: {4}         28: {1,1,4}       45: {2,2,3}
%e A335433      10: {1,3}       29: {10}          46: {1,9}
%e A335433      11: {5}         30: {1,2,3}       47: {15}
%e A335433      12: {1,1,2}     31: {11}          50: {1,3,3}
%e A335433      13: {6}         33: {2,5}         51: {2,7}
%e A335433      14: {1,4}       34: {1,7}         52: {1,1,6}
%e A335433      15: {2,3}       35: {3,4}         53: {16}
%e A335433      17: {7}         36: {1,1,2,2}     55: {3,5}
%e A335433      18: {1,2,2}     37: {12}          57: {2,8}
%e A335433      19: {8}         38: {1,8}         58: {1,10}
%t A335433 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A335433 Select[Range[100],Select[Permutations[primeMS[#]],!MatchQ[#,{___,x_,x_,___}]&]!={}&]
%Y A335433 The version for a multiset with prescribed multiplicities is A335127.
%Y A335433 Separable factorizations are counted by A335434.
%Y A335433 The complement is A335448.
%Y A335433 Separations are counted by A003242 and A335452 and ranked by A333489.
%Y A335433 Permutations of prime indices are counted by A008480.
%Y A335433 Inseparable partitions are counted by A325535.
%Y A335433 Strict permutations of prime indices are counted by A335489.
%Y A335433 Cf. A000961, A005117, A056239, A112798, A114938, A181796, A292884, A335407, A335451, A335516.
%K A335433 nonn
%O A335433 1,2
%A A335433 _Gus Wiseman_, Jul 02 2020