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Heinz numbers of double-free integer partitions.
40

%I #9 Jan 22 2019 17:05:48

%S 1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,20,22,23,25,26,27,28,29,31,

%T 32,33,34,35,37,38,39,40,41,43,44,45,46,47,49,50,51,52,53,55,56,57,58,

%U 59,61,62,64,67,68,69,70,71,73,74,75,76,77,79,80,81,82,83

%N Heinz numbers of double-free integer partitions.

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

%C An integer partition is double-free if no part is twice any other part.

%C Also numbers n such that if prime(m) divides n then prime(2m) does not divide n, i.e., numbers not divisible by any element of A319613.

%H Alois P. Heinz, <a href="/A320340/b320340.txt">Table of n, a(n) for n = 1..20000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Double-FreeSet.html">Double-Free Set</a>

%e The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (41), (32), (1111), (7), (8), (311), (51), (9), (33), (61), (222), (411).

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

%t Select[Range[100],Intersection[primeMS[#],2*primeMS[#]]=={}&]

%Y Cf. A018819, A056239, A087897, A101417, A112798, A120641, A276431, A305148, A319613, A323092, A323093.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 07 2019