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A185955 Fajtlowicz p-primes. 3
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 (list; graph; refs; listen; history; text; internal format)
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
KEYWORD
nonn
AUTHOR
Craig Eric Larson, Feb 07 2011
STATUS
approved

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Last modified April 13 12:42 EDT 2024. Contains 371641 sequences. (Running on oeis4.)