A322902 - OEIS

%I #7 Dec 31 2018 13:18:24

%S 1,2,3,4,5,7,8,9,11,13,16,17,19,21,23,25,27,29,31,32,37,41,43,47,49,

%T 53,57,59,61,63,64,67,71,73,79,81,83,89,97,101,103,107,109,113,115,

%U 121,125,127,128,131,133,137,139,147,149,151,157,159,163,167,169

%N Numbers whose prime indices are all proper powers of the same number.

%C A prime index of n is a number m such that prime(m) divides n.

%C A proper power of n is a number n^k for some positive integer k.

%C Also the union of A322903 and A000079.

%e The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (5), (6), (1111), (7), (8), (42), (9), (33), (222).

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

%t radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);

%t Select[Range[100],SameQ@@radbase/@primeMS[#]&]

%Y Cf. A001597, A018819, A023893, A023894, A052410, A056239, A072720, A072721,  A302242, A302593, A322900, A322901, A322903.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 30 2018