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 A323215 Numbers k such that row k of A322936 is not empty and has only primes as members. 1
 5, 8, 9, 10, 12, 18, 24, 30 (list; graph; refs; listen; history; text; internal format)
 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 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 Table of n, a(n) for n=1..8. Peter Luschny, Strong Coprimality. 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 *) 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 Cf. A181830, A141341, A307345, A322936, A322937. Sequence in context: A293271 A348046 A371656 * A334818 A337215 A314571 Adjacent sequences: A323212 A323213 A323214 * A323216 A323217 A323218 KEYWORD nonn,fini,full AUTHOR Peter Luschny, Apr 01 2019 EXTENSIONS Name corrected after a notice from Robert Israel by Peter Luschny, Apr 02 2019 STATUS approved

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Last modified September 12 10:44 EDT 2024. Contains 375850 sequences. (Running on oeis4.)