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A327823 Odd integers m such that every odd integer k with 1 < k < m and gcd(k,m) = 1 is prime.

%I A327823 #27 Oct 14 2019 14:26:14
%S A327823 1,3,5,7,9,15,21,45,105
%N A327823 Odd integers m such that every odd integer k with 1 < k < m and gcd(k,m) = 1 is prime.
%C A327823 Solomon W. Golomb and Kee-Wai Lau prove in AMM (see link) that the greatest odd integer with this property is 105.
%C A327823 This sequence is inspirated by the other one: integers q such that every integer k with 1 < k < q and gcd(k,q) = 1 is prime, with 2, 3, 4, 6, 8, 12, 18, 24, 30 in A048597 \ {1}.
%C A327823 The terms 1 and 3 are added after recommendations of _Amiram Eldar_ and _Michel Marcus_.
%D A327823 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, number 105, page 118.
%H A327823 Solomon W. Golomb and Kee-Wai Lau, <a href="https://www.jstor.org/stable/2322829">Problem E3137</a>, American Mathematical Monthly, Vol. 94, No. 9, Nov. 1987, pp. 883-884.
%e A327823 For m = 15 and 1 < k odd < 15, we have gcd(3,15) = 3, gcd(5,15) = 5, gcd(7,15) = 1, gcd(9,15) = 3, gcd(11,15) = 1, gcd(13,15) = 1. So, gcd(k,15) = 1 only if k is prime and 15 is a term.
%e A327823 For m = 63, we have gcd(25,63) = 1 with 25 no prime, so 63 is not a term.
%t A327823 aQ[n_] := OddQ[n] && AllTrue[Select[Range[3, n, 2], CoprimeQ[n, #] &], PrimeQ]; Select[Range[10^3], aQ] (* _Amiram Eldar_, Sep 27 2019 *)
%o A327823 (PARI) isok(m) = {if (m % 2, forstep (k=3, m-1, 2, if ((gcd(k, m) == 1) && !isprime(k), return(0));); return(1););} \\ _Michel Marcus_, Sep 27 2019
%Y A327823 Cf. A048597.
%K A327823 nonn,full,fini
%O A327823 1,2
%A A327823 _Bernard Schott_, Sep 26 2019