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A323215 Numbers k such that row k of A322936 is not empty and has only primes as members.

%I A323215 #39 Oct 18 2023 02:11:50
%S A323215 5,8,9,10,12,18,24,30
%N A323215 Numbers k such that row k of A322936 is not empty and has only primes as members.
%C A323215 a is strongly prime to n if and only if a <= n is prime to n and a does not divide n-1. See the link to 'Strong Coprimality'. (Our terminology follows the plea of Knuth, Graham and Patashnik in Concrete Mathematics, p. 115.)
%C A323215 From _Robert Israel_, Apr 02 2019: (Start)
%C A323215 If there is at least one prime <= sqrt(n) that divides neither n nor n-1, 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.
%C A323215 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)
%C A323215 There are no more terms.  See proof at A307345. - _Robert Israel_, Apr 03 2019
%H A323215 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/StrongCoprimality">Strong Coprimality</a>.
%p A323215 filter:= proc(n) local k, found;
%p A323215   found:= false;
%p A323215   for k from 2 to n-2 do
%p A323215     if igcd(k,n)=1 and (n-1) mod k <> 0 then
%p A323215       found:= true;
%p A323215       if not isprime(k) then return false fi;
%p A323215     fi
%p A323215   od;
%p A323215   found
%p A323215 end proc:
%p A323215 select(filter, [$1..1000]); # _Robert Israel_, Apr 02 2019
%t A323215 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 *)
%o A323215 (Sage) # uses[A322936row from A322936]
%o A323215 def isA323215(n):
%o A323215     return all(is_prime(p) for p in A322936row(n))
%o A323215 [n for n in (1..100) if isA323215(n)] # _Peter Luschny_, Apr 03 2019
%Y A323215 Cf. A181830, A141341, A307345, A322936, A322937.
%K A323215 nonn,fini,full
%O A323215 1,1
%A A323215 _Peter Luschny_, Apr 01 2019
%E A323215 Name corrected after a notice from _Robert Israel_ by _Peter Luschny_, Apr 02 2019