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

Author

Vladimir Shevelev, Sep 02 2009

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

Cf. A164333.

Programs

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 *)

Extensions

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