A185955 Fajtlowicz p-primes.

7, 23, 47, 167, 251, 359, 389, 839, 941, 1367, 1847, 1889, 2207, 2417, 3719, 3761, 4889, 5039, 6311, 7079, 7919, 8609, 9377, 10607, 11411, 11447, 13841, 15227, 16127, 17159, 18869, 19319, 20411, 24611, 25589, 26669, 29501, 29927, 36017, 36479, 37907, 43037, 44519, 44927, 45569, 49727, 50627, 52889, 54287, 57119, 62057, 65309, 66047, 70529, 85037, 85847, 95369, 97967, 99191

Offset

1,1

Comments

S. Fajtlowicz defined two related sequences of primes, p(n) and q(n), as follows:

1. q(1)=2 and p(1)=7.

2. q(n+1) is the smallest prime dividing p(n)+2.

3. p(n+1) is the smallest prime p larger than p(n) such that p+2 is not prime and not divisible by any of q(1),q(2),...,q(n+1).

Paul Erdős proved that the series of reciprocal of the p-primes converges.

The values of p and q were computed by Bethany Turner.

References

Siemion Fajtlowicz, Written on the Wall: Conjectures of Graffiti, #784 (1994).

Links

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..201 from R. J. Mathar)

Siemion Fajtlowicz, Graffity & automated conjecture making (2009), click on "conjectures up to No. 894", see page 136.

Maple

A185955 := proc(n)
    option remember;
    local a, admit, k ;
    if n = 1 then
        7;
    else
        a := ithprime(n) ;
        while true do
            if not isprime(a+2) then
                admit := true ;
                for k from 1 to n do
                    if modp(a+2, A185956(k)) =0 then
                        admit := false;
                        break;
                    end if;
                end do:
                if admit then
                    return a;
                end if ;
            end if;
            a := nextprime(a) ;
        end do:
    end if;
end proc ;
seq(A185955(n), n=1..20) ; # R. J. Mathar, Jul 28 2019

Mathematica

lpf[n_] := FactorInteger[n][[1, 1]]; q[1] = 2; p[1] = 7; q[n_] := q[n] = lpf[p[n - 1] + 2]; p[n_] := Module[{pn = NextPrime[p[n - 1]]}, While[PrimeQ[pn + 2] || AnyTrue[Array[q, n], Divisible[pn + 2, #] &], pn = NextPrime[pn]]; pn]; Array[p, 50] (* Amiram Eldar, Apr 23 2021 *)

Crossrefs

Cf. A185956.

Sequence in context: A348230 A139830 A153210 * A158035 A101789 A174590

Adjacent sequences: A185952 A185953 A185954 * A185956 A185957 A185958

Keyword

nonn

Author

Craig Eric Larson, Feb 07 2011

Status

approved