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A225512 -8-Knödel numbers.

%I A225512 #22 Feb 16 2025 08:33:19
%S A225512 4,6,8,12,16,20,22,24,28,32,40,48,52,60,80,96,112,120,132,136,160,208,
%T A225512 240,280,322,352,364,408,480,520,532,580,680,682,952,1036,1120,1312,
%U A225512 1392,1456,1612,1768,1840,2040,2080,2332,2584,3016,3172,3268,3472,3640
%N A225512 -8-Knödel numbers.
%C A225512 Extension of k-Knodel numbers to k negative, in this case equal to -8. Composite numbers n > 0 such that if 1 < a < n and gcd(n,a) = 1 then a^(n+8) = 1 mod n.
%H A225512 Rémy Sigrist, <a href="/A225512/b225512.txt">Table of n, a(n) for n = 1..1000</a>
%H A225512 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KnoedelNumbers.html">Knödel Numbers</a>
%p A225512 with(numtheory); ListA225512:=proc(q,k) local a,n,ok;
%p A225512 for n from 2 to q do if not isprime(n) then ok:=1; for a from 1 to n do
%p A225512 if gcd(a,n)=1 then if (a^(n-k)-1) mod n<>0 then ok:=0; break; fi; fi;
%p A225512 od; if ok=1 then print(n); fi; fi; od; end: ListA225512(10^6,-8);
%t A225512 Select[Range[10000], CompositeQ[#] && Divisible[# + 8, CarmichaelLambda[#]] &] (* _Amiram Eldar_, Mar 28 2019 *)
%o A225512 (PARI) is(n) = if (bigomega(n)>1, for (a=2, n-1, if (gcd(n,a)==1 && Mod(a,n)^(n+8)!=1, return (0))); return (1), return (0)) \\ _Rémy Sigrist_, Mar 03 2019
%Y A225512 Cf. A208728.
%Y A225512 Cf. A225506, A225507, A225508, A225509, A225510, A225511, A225513, A225514.
%K A225512 nonn
%O A225512 1,1
%A A225512 _Paolo P. Lava_, May 09 2013
%E A225512 More terms from _Rémy Sigrist_, Mar 03 2019