A322531 - OEIS

%I #7 Dec 14 2018 21:21:05

%S 1,2,3,4,5,8,11,13,15,16,17,29,31,32,33,41,43,47,51,55,59,64,67,73,79,

%T 83,85,93,101,109,113,123,127,128,137,139,149,155,157,163,165,167,177,

%U 179,181,187,191,199,201,205,211,233,241,249,255,256,257,269,271

%N Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%C All entries are themselves squarefree numbers (except the powers of 2).

%C The first odd term not in this sequence but in A302521 is 141, which is the MM-number (see A302242) of {{1},{2,3}}.

%e The sequence of all integer partitions whose parts all have the same number of prime factors and whose product of parts is a squarefree number begins: (), (1), (2), (1,1), (3), (1,1,1), (5), (6), (3,2), (1,1,1,1), (7), (10), (11), (1,1,1,1,1), (5,2), (13), (14), (15), (7,2), (5,3), (17), (1,1,1,1,1,1).

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

%t Select[Range[100],And[SameQ@@PrimeOmega/@primeMS[#],SquareFreeQ[Times@@primeMS[#]]]&]

%Y Cf. A003963, A005117, A038041, A056239, A073576, A112798, A302242, A302505, A306017, A319056, A319169, A320324, A321717, A321718, A322526, A322528.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 14 2018