1,2

Apparently A136119(n) = A001953(n-1)+1 = floor((n-1/2)sqrt(2))+1 (confirmed for n < 20000), and A136119(n+1) - A136119(n) = A001030(n). From the definitions these conjectures are by no means obvious. Can they be proved? (Klaus Brockhaus, Apr 15 2008) The answer seems to be Yes - see the Cloitre link.

D. X. Charles : Sieve Methods, July 2000, University of Wisconsin. http://pages.cs.wisc.edu/~cdx/Sieve.pdf

B. Cloitre, The golden sieve, preprint 2008

R. Eismann : Decomposition of natural numbers into weight X level + jump and application to a new classification of prime numbers, ArXiv 2008. http://arXiv.org/PS_cache/arXiv/pdf/0711/0711.0865v2.pdf

M. C. Wunderlich : A general class of sieve generated sequences, Acta Arithmetica XVI,1969, pp.41-56. http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1614.pdf

Benoit Cloitre, On the proof of Klaus Brockhaus's conjectures

Index entries for sequences generated by sieves

G.f.: (-x^9 + x^8 + 2*x^4 + x^3 + x^2 + 2*x + 1)/(x^6 - x^5 - x + 1)

a(n)=ceil((n-1/2)sqrt(2)). This can be proved in the same way as the formula given for A099267. There are some generalisations. For instance, it is possible to consider "a(n)+K*n" instead of "a(n)+n" for deleting terms where K=0,1,2,...is fixed. The constant involved in the Beatty sequence for the sequence of deleted terms then depends on K and equals (K+1+sqrt((K+1)^2+4))/2. K=0 is related to A099267. 1+A001954 is the complement sequence of this sequence A136119. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 18 2008

First few steps are:

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...

n = 1; delete term at position 1+a(1) = 2: 2;

1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...

n = 2; delete term at position 2+a(2) = 5: 6;

1,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...

n = 3; delete term at position 3+a(3) = 7: 9;

1,3,4,5,7,8,10,11,12,13,14,15,16,17,18,19,20,...

n = 4; delete term at position 4+a(4) = 9: 12;

1,3,4,5,7,8,10,11,13,14,15,16,17,18,19,20,...

n = 5; delete term at position 5+a(5) = 12: 16;

1,3,4,5,7,8,10,11,13,14,15,17,18,19,20,...

n = 6; delete term at position 6+a(6) = 14: 19;

1,3,4,5,7,8,10,11,13,14,15,17,18,20,...

Cf. A000027, A001953 (floor((n+1/2)*sqrt(2))), A001030 (fixed under 1 -> 21, 2 -> 211), A136110, A137292.

Cf. A000959, A099267.

Adjacent sequences: A136116 A136117 A136118 this_sequence A136120 A136121 A136122

Sequence in context: A047367 A039043 A116591 this_sequence A110882 A089230 A098090

easy,nonn

Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), Mar 16 2008

Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 15 2008