%I #6 Mar 18 2019 08:14:40

%S 1,3,5,7,9,11,13,17,19,21,23,25,27,29,31,33,35,37,41,43,47,49,51,53,

%T 57,59,61,63,65,67,71,73,77,79,81,83,85,87,89,91,93,95,97,99,101,103,

%U 107,109,113,115,121,123,125,127,129,131,133,137,139,143,147,149

%N Heinz numbers of integer partitions not containing 1 or any prime indices of the parts.

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

%e The sequence of terms together with their prime indices begins:

%e 1: {}

%e 3: {2}

%e 5: {3}

%e 7: {4}

%e 9: {2,2}

%e 11: {5}

%e 13: {6}

%e 17: {7}

%e 19: {8}

%e 21: {2,4}

%e 23: {9}

%e 25: {3,3}

%e 27: {2,2,2}

%e 29: {10}

%e 31: {11}

%e 33: {2,5}

%e 35: {3,4}

%e 37: {12}

%e 41: {13}

%e 43: {14}

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

%t Select[Range[1,100,2],Intersection[primeMS[#],Union@@primeMS/@primeMS[#]]=={}&]

%Y The subset version is A324742, with maximal case A324763. The strict integer partition version is A324752. The integer partition version is A324757. An infinite version is A324695.

%Y Cf. A000720, A001221, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324743, A324751, A324756, A324758, A324764.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 17 2019