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A305732 Heinz numbers of reducible integer partitions. Numbers n > 1 that are prime or whose prime indices are relatively prime and such that A181819(n) is already in the sequence.

%I A305732 #10 Jun 23 2018 14:20:55
%S A305732 2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,18,19,20,22,23,24,26,28,29,30,
%T A305732 31,32,33,34,35,37,38,40,41,42,43,44,45,46,47,48,50,51,52,53,54,55,56,
%U A305732 58,59,60,61,62,64,66,67,68,69,70,71,72,73,74,75,76,77,78
%N A305732 Heinz numbers of reducible integer partitions. Numbers n > 1 that are prime or whose prime indices are relatively prime and such that A181819(n) is already in the sequence.
%C A305732 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A prime index of n is a number m such that prime(m) divides n. A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is reducible if either m is of size 1 or gcd(m_1,...,m_k) = 1 and the multiset {y_1,...,y_k} is also reducible.
%e A305732 60 has relatively prime prime indices {1,1,2,3} with multiplicities {1,1,2} corresponding to A181819(90) = 12. 12 has relatively prime prime indices {1,1,2} with multiplicities {1,2} corresponding to A181819(12) = 6. 6 has relatively prime prime indices {1,2} with multiplicities {1,1} corresponding to A181819(6) = 4. 4 has relatively prime prime indices {1,1} with multiplicities {2} corresponding to A181819(4) = 3. 3 is prime, so we conclude that 60 belongs to the sequence.
%t A305732 rdzQ[n_]:=And[n>1,Or[PrimeQ[n],And[rdzQ[Times@@Prime/@FactorInteger[n][[All,2]]],GCD@@PrimePi/@FactorInteger[n][[All,1]]==1]]];
%t A305732 Select[Range[50],rdzQ]
%Y A305732 Cf. A000837, A007916, A056239, A181819, A182850, A289508, A289509, A298748, A304465, A304687, A304818, A305563, A305731, A305733.
%K A305732 nonn
%O A305732 1,1
%A A305732 _Gus Wiseman_, Jun 22 2018