0,3

Comment from David Newman, Feb 17 2009: (Start) This sequence also arises in the following way.

Call a set A of nonnegative integers a basis if every nonnegative integer can be written as the sum of two (not necessarily distinct) elements of A.

Call a basis an increasing basis if its elements are arranged in increasing order, a0< a1< a2<...

For example A126684 : 0, 1, 2, 4, 5, 8, 10, 16, 17, 20, 21, 32, 34, 40,... is an increasing basis.

Now consider the set of all initial subsequences of any length {a0, a1, a2,...,an} of all the increasing bases. These can be ordered in the library ordering, giving:

0

0, 1

0, 1, 2

0, 1, 3

0, 1, 2, 3

0, 1, 2, 4

0, 1, 2, 5

0, 1, 3, 4

0, 1, 3, 5

...

How many such subsequences are there of length n? The answer is a(n+1).

A Goldbach sequence is then an increasing basis without the initial zero. (End)

Comment from Martin Fuller: The largest value for each term in any increasing basis is given by A123509.

M. Torelli, Increasing integer sequences and Goldbach's conjecture, preprint, 1996.

Index entries for sequences related to Goldbach conjecture

(PARI code from Martin Fuller) A008932(n, pol=0)= { local(a=0, i, pol2);

!n & return(1);

i = #pol;

pol2 = pol^2;

for (i=#pol, #pol2+1,

a += A008932(n-1, pol+'x^i);

!polcoeff(pol2, i) & break; );

a }

Cf. A123509.

Sequence in context: A150013 A123166 A052539 this_sequence A062881 A122206 A104082

Adjacent sequences: A008929 A008930 A008931 this_sequence A008933 A008934 A008935

nonn,more

torelli(AT)hermes.mc.dsi.unimi.it (Mauro Torelli)

a(9) - a(14) from Martin Fuller (martin_n_fuller(AT)btinternet.com), Feb 18 2009

Edited by N. J. A. Sloane, Mar 12 2009