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%I #22 Jan 17 2021 03:14:01
%S 35,1189,3059,6479,8569,30889,39689,51271,84419,133399,272251,321265,
%T 430199,544159,564719,569449,585311,608399,1033241,1212751,1930499,
%U 3131029,7056721,7110179,7639919,8740601,11255201,15857855,17966519,18996769,22427999,32871761,34966009
%N Composite squarefree numbers k such that k^2-1 is divisible by p-1 and p+1, where p are all the prime factors of k.
%H Amiram Eldar, <a href="/A306685/b306685.txt">Table of n, a(n) for n = 1..100</a>
%e Prime factors of 35 are 5, 7 and 35^2-1 = 1224, 1124/4 = 306, 1124/6 = 204, 1124/8 = 153.
%e Prime factors of 1189 are 29, 41 and 1189^2-1 = 1413720, 1413720/28 = 50490, 1413720/30 = 47124, 1413720/40 = 35343, 1413720/42 = 33660.
%p with(numtheory): P:=proc(q) local a,k,ok,n; for n from 2 to q do
%p if not isprime(n) and issqrfree(n) then a:=factorset(n); ok:=1;
%p for k from 1 to nops(a) do if frac((n^2-1)/(a[k]+1))>0 or frac((n^2-1)/(a[k]-1))>0 then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^9);
%t csfQ[n_]:=CompositeQ[n]&&SquareFreeQ[n]&&Union[Mod[n^2-1,Flatten[{#+1, #-1}&/@ FactorInteger[n][[All,1]]]]]=={0}; Select[Range[35*10^6],csfQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Apr 07 2020 *)
%o (PARI) isok(n) = {if (issquarefree(n) && !isprime(n) && (n>1), my(f = factor(n)[,1], x = n^2-1); for (k=1, #f, if ((x % (f[k]-1)) || (x % (f[k]+1)), return (0));); return (1);); return (0);} \\ _Michel Marcus_, Mar 12 2019
%Y Cf. A002997, A006972, A208728, A225711, A304291, A306723.
%K nonn
%O 1,1
%A _Paolo P. Lava_, Mar 05 2019
%E More terms from _Giovanni Resta_, Mar 06 2019
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