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

5, 8, 9, 10, 12, 18, 24, 30

Offset: 1

Author

Peter Luschny, Apr 01 2019

Keywords

• nonn
• fini
• full

Comments

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.)

From Robert Israel, Apr 02 2019: (Start)

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.

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)

There are no more terms. See proof at A307345. - Robert Israel, Apr 03 2019

Links

• Peter Luschny, Strong Coprimality.

Crossrefs

Cf. A181830, A141341, A307345, A322936, A322937.

Programs

Maple

filter:= proc(n) local k, found;
  found:= false;
  for k from 2 to n-2 do
    if igcd(k,n)=1 and (n-1) mod k <> 0 then
      found:= true;
      if not isprime(k) then return false fi;
    fi
  od;
  found
end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 02 2019

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 *)

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

Extensions

Name corrected after a notice from Robert Israel by Peter Luschny, Apr 02 2019