A164960 The minimum number of steps needed to generate prime(n) under the map x -> A060264(x) starting from any x taken from {2,3} or from A164333.

0, 0, 1, 1, 2, 0, 2, 0, 3, 1, 0, 3, 1, 0, 4, 0, 2, 0, 1, 0, 0, 4, 2, 1, 5, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 5, 3, 0, 2, 0, 0, 0, 6, 0, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 3, 1, 0, 0, 1, 0, 0, 1, 6, 4, 1, 0, 0, 3, 1, 0, 0, 1, 1, 7, 1

Offset

1,5

Links

Table of n, a(n) for n=1..79.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Example

a(3) = 1 because prime(3)=5 can be generated in 1 step starting from x=2.
a(4) = 1 because prime(4)=7 can be generated in 1 step starting from x=3.

Maple

# include source from A164333 and A060264 here
A164333 := proc(n)
        if n = 1 then
                13;
        else
                for a from procname(n-1)+1 do
                        if isA164333(a) then
                                return a;
                        end if;
                end do;
        end if;
end proc:
A164960aux := proc(p, strt)
        local a, x;
        if strt > p then
                return 1000000000;
        end if;
        a := 0 ;
        x := strt ;
        while x < p do
                x := A060264(x) ;
                a := a+1 ;
        end do;
        if x = p then
                return a ;
        else
                return 1000000000;
        end if;
end proc:
A164960 := proc(n)
        local p, a, strt, i;
        p := ithprime(n) ;
        a := A164960aux(p, 2) ;
        a := min(a, A164960aux(p, 3)) ;
        for i from 1 do
                strt := A164333(i) ;
                if strt > p then
                        return a;
                else
                        a := min(a, A164960aux(p, strt)) ;
                end if;
        end do:
        return a;
end proc:
seq(A164960(n), n=1..90) ; # R. J. Mathar, Oct 29 2011

Mathematica

nmax = 100; kmax = nmax + 5;
A164333 = Select[Table[{(Prime[k - 1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, kmax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ] &][[All, 2]]*2 + 1;
A164960aux[p_, strt_] := Module[{a, x}, If[strt > p, Return[10^9]]; a = 0; x = strt; While[x < p, x = NextPrime[2 x]; a++]; If[x == p, Return[a], Return[10^9]]];
A164960[n_] := Module[{p, a, strt, i}, p = Prime[n]; a = A164960aux[p, 2]; a = Min[a, A164960aux[p, 3]]; For[i = 1, i < 100, i++, strt = A164333[[i]]; If[strt > p, Return[a], a = Min[a, A164960aux[p, strt]]]]; Return[a]];
Table[A164960[n], {n, 1, nmax}] (* Jean-François Alcover, Dec 13 2017, after R. J. Mathar *)

Crossrefs

Cf. A164333.

Sequence in context: A326844 A261079 A182485 * A124137 A164092 A302643

Adjacent sequences: A164957 A164958 A164959 * A164961 A164962 A164963

Keyword

nonn

Author

Vladimir Shevelev, Sep 02 2009

Extensions

One term corrected, sequence extended, examples added by R. J. Mathar, Oct 29 2011

Status

approved