login
Heinz numbers of integer partitions where the set of distinct parts is disjoint from the set of distinct multiplicities.
24

%I #5 Apr 01 2019 07:25:06

%S 1,3,4,5,7,8,11,13,15,16,17,19,21,23,25,27,29,31,32,33,35,37,39,41,43,

%T 47,49,51,53,55,57,59,61,64,65,67,69,71,73,77,79,81,83,85,87,89,91,93,

%U 95,97,100,101,103,105,107,109,111,113,115,119,121,123,127

%N Heinz numbers of integer partitions where the set of distinct parts is disjoint from the set of distinct multiplicities.

%C The enumeration of these partitions by sum is given by A114639.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers where the prime indices are disjoint from the prime exponents.

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

%e 1: {}

%e 3: {2}

%e 4: {1,1}

%e 5: {3}

%e 7: {4}

%e 8: {1,1,1}

%e 11: {5}

%e 13: {6}

%e 15: {2,3}

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

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

%e 33: {2,5}

%t Select[Range[100],Intersection[PrimePi/@First/@FactorInteger[#],Last/@FactorInteger[#]]=={}&]

%Y Cf. A000720, A001222, A056239, A109298, A112798, A114639, A118914.

%Y Cf. A324571, A325127, A325128, A325129, A325130.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 01 2019