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%I #8 Jun 25 2021 23:41:26
%S 1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,19,20,21,22,24,26,28,29,30,
%T 31,32,33,34,35,37,38,39,40,41,42,43,44,47,48,51,52,53,55,56,57,58,59,
%U 60,62,64,65,66,67,68,70,71,73,74,76,77,78,79,80,82,84,85
%N Matula-Goebel numbers of rooted semi-identity trees.
%C Definition: A positive integer belongs to the sequence iff its prime indices greater than 1 are distinct and already belong to the sequence. 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.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%e The sequence of all unlabeled rooted semi-identity trees together with their Matula-Goebel numbers begins:
%e 1: o
%e 2: (o)
%e 3: ((o))
%e 4: (oo)
%e 5: (((o)))
%e 6: (o(o))
%e 7: ((oo))
%e 8: (ooo)
%e 10: (o((o)))
%e 11: ((((o))))
%e 12: (oo(o))
%e 13: ((o(o)))
%e 14: (o(oo))
%e 15: ((o)((o)))
%e 16: (oooo)
%e 17: (((oo)))
%e 19: ((ooo))
%e 20: (oo((o)))
%e 21: ((o)(oo))
%e 22: (o(((o))))
%e 24: (ooo(o))
%e 26: (o(o(o)))
%e 28: (oo(oo))
%e 29: ((o((o))))
%e 30: (o(o)((o)))
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t psidQ[n_]:=And[UnsameQ@@DeleteCases[primeMS[n],1],And@@psidQ/@primeMS[n]];
%t Select[Range[100],psidQ]
%Y Cf. A000081, A004111, A007097, A276625, A277098, A306200, A306201, A316467.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 29 2019
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