A303945 - OEIS

%I #5 May 06 2018 12:43:25

%S 1,2,3,1,2,4,1,3,5,1,1,2,6,1,4,2,3,7,1,2,2,8,1,1,3,2,4,1,5,9,1,1,1,2,

%T 1,6,1,1,4,10,1,2,3,11,2,5,1,7,3,4,12,1,8,2,6,1,1,1,3,13,1,2,4,14,1,1,

%U 5,2,2,3,1,9,15,1,1,1,1,2,1,3,3,2,7,1,1

%N Triangle whose n-th row lists the multiset of prime indices of the n-th number that is not a perfect power A007916(n).

%C A number is not a perfect power if its prime multiplicities are relatively prime. This triangle is an enumeration of all finite aperiodic multisets of positive integers, where a multiset is aperiodic if its multiplicities are relatively prime.

%e Sequence of numbers that are not perfect powers together with their multisets of prime indices begins:

%e 2:  {1}

%e 3:  {2}

%e 5:  {3}

%e 6:  {1,2}

%e 7:  {4}

%e 10: {1,3}

%e 11: {5}

%e 12: {1,1,2}

%e 13: {6}

%e 14: {1,4}

%e 15: {2,3}

%e 17: {7}

%e 18: {1,2,2}

%e 19: {8}

%e 20: {1,1,3}

%e 21: {2,4}

%e 22: {1,5}

%e 23: {9}

%e 24: {1,1,1,2}

%e 26: {1,6}

%e 28: {1,1,4}

%e 29: {10}

%e 30: {1,2,3}

%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t primeMS/@Select[Range[2,100],GCD@@FactorInteger[#][[All,2]]===1&]

%Y Cf. A000041, A000720, A000837, A001222, A005117, A007916, A052409, A052410, A056239, A112798, A296150.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, May 03 2018