Duplicated sequences.

N. J. A. Sloane njas at research.att.com
Thu Feb 15 16:43:28 CET 2007

so here:
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Date: Thu, 15 Feb 2007 17:35:13 +0100
From: reismann at free.fr
To: seqfan at ext.jussieu.fr
Subject: COMMENT on A006562 and A001359 (twin and balanced primes)
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Dear Neil, Seqfans,

I sent these two comments :

- On A001359 : Lesser of twin primes
%I A001359
%C A001359 Lesser of twin primes - (3) = Primes for which the weight as defined
in A117078 is 3.
%O A001359 0
%K A001359 ,nonn,
%A A001359 Remi Eismann (reismann at free.fr), Feb 15 2007

- and on A006562 : Balanced primes
%C A006562 Let p(i) denote the i-th prime. If 2 p(n) - p(n+1) is a prime, say
p(n-i) and if p(n) has a level 1 in A117563, then we say that p(n) has
level(1,i). Sequence gives primes of level(1,1).
%O A006562 0
%K A006562 ,nonn,
%A A006562 Remi Eismann (reismann at free.fr), Feb 15 2007



%I A124593
%S A124593 1,1,1,2,3,6,11,13
%N A124593 Number of 4-indecomposable trees with n nodes.
%C A124593 A connected graph is called k-decomposable if it is possible to remove some edges and leave a graph with at least two connected components in which every component has at least k nodes.
%C A124593 Every connected graph with < 2k nodes is automatically k-indecomposable.
%C A124593 A 4-indecomposable tree may not contain a path with >= 8 nodes, nor two node-disjoint paths with >= 4 nodes each.
%C A124593 The counts of 1-indecomposable (1,0,0,0,...), 2-indecomposable (1,1,1,1,1,1,...) or 3-indecomposable (1,1,1,2,3,3,4,4,5,5,6,6,7,7,...) trees are all trivial.
%e A124593 Rather than show some 4-indecomposable trees, instead we give all four 3-indecomposable trees with 7 nodes:
%e A124593 O-O-O-O-O....O..........O.O...O...O
%e A124593 ....|........|..........|/.....\./.
%e A124593 ....O....O-O-O-O-O..O-O-O-O...O-O-O
%e A124593 ....|........|..........|....../.\.
%e A124593 ....O........O..........O.....O...O
%e A124593 On the other hand, O-O-O-O-O-O-O is 3-decomposable, because removing the third edge gives O-O-O O-O-O-O, with 2 connected components each with >= 3 nodes.
%Y A124593 Cf. A000055, A125709.
%K A124593 nonn,more,new
%O A124593 1,4
%A A124593 David Applegate and njas, Feb 14 2007
%E A124593 a(8) needs to be checked and the sequence extended. At present this entry is just a place-holder. It would also be good to get the analogous sequences for 5- and 6-indecomposable trees, as well as those in which the degree of every node is at most 4. The same questions can be asked about connected graphs rather than trees.

Some nice programming challenges!


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