A324758 - OEIS

%I #6 Mar 18 2019 08:14:17

%S 1,2,3,4,5,7,8,9,10,11,13,16,17,19,20,21,22,23,25,27,29,31,32,33,34,

%T 35,37,40,41,43,44,46,47,49,50,51,53,57,59,61,62,63,64,65,67,68,71,73,

%U 77,79,80,81,82,83,85,87,88,89,91,92,93,94,95,97,99,100,101

%N Heinz numbers of integer partitions containing no 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).

%C These could be described as anti-transitive numbers (cf. A290822), as they are numbers x such that if prime(y) divides x and prime(z) divides y, then prime(z) does not divide x.

%C Also numbers n such that A003963(n) is coprime to n.

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

%e    1: {}

%e    2: {1}

%e    3: {2}

%e    4: {1,1}

%e    5: {3}

%e    7: {4}

%e    8: {1,1,1}

%e    9: {2,2}

%e   10: {1,3}

%e   11: {5}

%e   13: {6}

%e   16: {1,1,1,1}

%e   17: {7}

%e   19: {8}

%e   20: {1,1,3}

%e   21: {2,4}

%e   22: {1,5}

%e   23: {9}

%e   25: {3,3}

%e   27: {2,2,2}

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

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

%Y The subset version is A324741, with maximal case A324743. The strict integer partition version is A324751. The integer partition version is A324756. An infinite version is A324695.

%Y Cf. A000720, A001221, A001462, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324742, A324753, A324764.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 17 2019