1,1

Theorem. For every m>4 there exists n=n(m) such that the minimal recursive sequence beginning with m similar to N with respect to the considered property coincides with A159619 for n>=n(m). Thus there exist essentially two minimal recursive sequences similar to N with respect to such property: A159615 and A159619.

In connection with this theorem, consider the following problems. For a given increasing sequence of positive integers {c(n)} (n>=1) and for a>c(1), denote {c_a(n)} the minimal recursive sequence beginning with a, which is similar to N with respect to the following property of an integer: to be or not to be in {c(n)}. We call rank of {c(n)} the least number r such that, for every a>r, for all sufficiently large n>=n(a), we have c_a(n)=c_r(n). In particular, if c(n)=A004760(n+1), then {c(n)} has rank r=c_2, while if c(n)=A000069(n), then {c(n)}has rank r=c_3. The problems are: 1)to find a sequence of rank r>=c_4; 2) to find rhe rank of primes or to prove that it does not exist (in this case one can say that r equals to infinity). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 23 2009]

V. Shevelev, Several results on sequences which are similar to the positive integers

a(n)=2n+3, if n*A007814(n+1) is even and a(n)=2n+2, otherwise.

A159615 A007814 A004760 A159559 A159560

Sequence in context: A010454 A053169 A007656 this_sequence A131827 A035245 A109180

Adjacent sequences: A159616 A159617 A159618 this_sequence A159620 A159621 A159622

nonn,uned

Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 17 2009, Apr 27 2009, May 04 2009