

A323215


Numbers k such that row k of A322936 is not empty and has only primes as members.


1




OFFSET

1,1


COMMENTS

a is strongly prime to n if and only if a <= n is prime to n and a does not divide n1. See the link to 'Strong Coprimality'. (Our terminology follows the plea of Knuth, Graham and Patashnik in Concrete Mathematics, p. 115.)
If there is at least one prime <= sqrt(n) that divides neither n nor n1, then its square is strongly prime to n and not prime. If there does not exist such a prime, then the first Chebyshev function theta(sqrt(n)) = Sum_{p <= sqrt(n)} log(p) <= 2 log(n). Now it is known that theta(x) = x + O(x/log(x)), so this can't happen if n is sufficiently large. Thus the sequence is finite.
The largest n for which no such p exists appears to be 120. There are none between 121 and 10^7. It is possible that a sufficiently tight lower bound on theta together with a finite search can be used to prove that there are no other terms of the sequence. (End)


LINKS



MAPLE

filter:= proc(n) local k, found;
found:= false;
for k from 2 to n2 do
if igcd(k, n)=1 and (n1) mod k <> 0 then
found:= true;
if not isprime(k) then return false fi;
fi
od;
found
end proc:


MATHEMATICA

Select[Range[10^3], With[{n = #}, AllTrue[Select[Range[2, n], And[GCD[#, n] == 1, Mod[n  1, #] != 0] &] /. {} > {0}, PrimeQ]] &] (* Michael De Vlieger, Apr 01 2019 *)


PROG

(Sage) # uses[A322936row from A322936]
def isA323215(n):
return all(is_prime(p) for p in A322936row(n))
[n for n in (1..100) if isA323215(n)] # Peter Luschny, Apr 03 2019


CROSSREFS



KEYWORD

nonn,fini,full


AUTHOR



EXTENSIONS



STATUS

approved



