cp's OEIS Frontend

A331683 One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.

%I A331683 #14 Feb 06 2020 01:04:28
%S A331683 1,4,8,14,16,28,32,38,56,64,76,86,106,112,128,152,172,212,214,224,256,
%T A331683 262,304,326,344,424,428,448,512,524,526,608,622,652,688,766,848,856,
%U A331683 886,896,1024,1048,1052,1154,1216,1226,1244,1304,1376,1438,1532,1696
%N A331683 One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.
%C A331683 Also Matula-Goebel numbers of lone-child-avoiding rooted trees at with at most one non-leaf branch under any given vertex. A rooted tree is lone-child-avoiding if there are no unary branchings. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of the root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
%C A331683 Also Matula-Goebel numbers of lone-child-avoiding locally disjoint semi-identity trees. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct.
%H A331683 Robert Israel, <a href="/A331683/b331683.txt">Table of n, a(n) for n = 1..3140</a>
%F A331683 Intersection of A291636, A316495, and A306202.
%e A331683 The sequence of all lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex together with their Matula-Goebel numbers begins:
%e A331683     1: o
%e A331683     4: (oo)
%e A331683     8: (ooo)
%e A331683    14: (o(oo))
%e A331683    16: (oooo)
%e A331683    28: (oo(oo))
%e A331683    32: (ooooo)
%e A331683    38: (o(ooo))
%e A331683    56: (ooo(oo))
%e A331683    64: (oooooo)
%e A331683    76: (oo(ooo))
%e A331683    86: (o(o(oo)))
%e A331683   106: (o(oooo))
%e A331683   112: (oooo(oo))
%e A331683   128: (ooooooo)
%e A331683   152: (ooo(ooo))
%e A331683   172: (oo(o(oo)))
%e A331683   212: (oo(oooo))
%e A331683   214: (o(oo(oo)))
%e A331683   224: (ooooo(oo))
%p A331683 N:= 10^4: # for terms <= N
%p A331683 S:= {1}:
%p A331683 with(queue):
%p A331683 Q:= new(1):
%p A331683 while not empty(Q) do
%p A331683   r:= dequeue(Q);
%p A331683   p:= ithprime(r);
%p A331683   newS:= {seq(2^i*p,i=1..ilog2(N/p))} minus S;
%p A331683   S:= S union newS;
%p A331683   for s in newS do enqueue(Q,s) od:
%p A331683 od:
%p A331683 sort(convert(S,list)); # _Robert Israel_, Feb 05 2020
%t A331683 uryQ[n_]:=n==1||MatchQ[FactorInteger[n],({{2,_},{p_,1}}/;uryQ[PrimePi[p]])|({{2,k_}}/;k>1)];
%t A331683 Select[Range[100],uryQ]
%Y A331683 These trees counted by number of vertices are A212804.
%Y A331683 The semi-lone-child-avoiding version is A331681.
%Y A331683 The non-semi-identity version is A331871.
%Y A331683 Lone-child-avoiding rooted trees are counted by A001678.
%Y A331683 Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
%Y A331683 Unlabeled semi-identity trees are counted by A306200, with Matula-Goebel numbers A306202.
%Y A331683 Locally disjoint rooted trees are counted by A316473.
%Y A331683 Matula-Goebel numbers of locally disjoint rooted trees are A316495.
%Y A331683 Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
%Y A331683 Cf. A000081, A007097, A061775, A196050, A276625, A316470, A316696, A331678, A331679, A331680, A331682, A331686, A331687.
%K A331683 nonn
%O A331683 1,2
%A A331683 _Gus Wiseman_, Jan 30 2020