A329366 - OEIS

%I #5 Nov 12 2019 19:24:15

%S 1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43,47,49,53,

%T 59,61,64,67,71,73,79,81,83,89,91,97,101,103,107,109,113,121,125,127,

%U 128,131,137,139,149,151,157,163,167,169,173,179,181,191,193,197

%N Numbers whose distinct prime indices are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

%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.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are Heinz numbers of stable intersecting partitions.

%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    11: {5}

%e    13: {6}

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

%e    17: {7}

%e    19: {8}

%e    23: {9}

%e    25: {3,3}

%e    27: {2,2,2}

%e    29: {10}

%e    31: {11}

%e    32: {1,1,1,1,1}

%e    37: {12}

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

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t Select[Range[100],stableQ[Union[primeMS[#]],GCD[#1,#2]==1&]&&stableQ[Union[primeMS[#]],Divisible]&]

%Y Intersection of A316476 and A328867.

%Y Heinz numbers of the partitions counted by A328871.

%Y Replacing "intersecting" with "relatively prime" gives A328677.

%Y Cf. A056239, A112798, A285573, A289509, A303362, A304713, A327393, A328671.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 12 2019