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A349546 Composite numbers k such that k+1 is divisible by (k+1 mod A001414(k)) and k-1 is divisible by (k-1 mod A001414(k)).

%I A349546 #14 Sep 29 2024 08:56:14
%S A349546 4,8,20,32,50,55,64,77,80,98,110,115,125,128,152,170,216,242,243,256,
%T A349546 275,290,329,338,341,343,364,371,416,506,511,512,544,551,578,583,611,
%U A349546 638,663,722,729,731,741,851,870,920,987,1024,1025,1054,1058,1079,1144,1219,1243,1298,1325,1331,1421
%N A349546 Composite numbers k such that k+1 is divisible by (k+1 mod A001414(k)) and k-1 is divisible by (k-1 mod A001414(k)).
%H A349546 Robert Israel, <a href="/A349546/b349546.txt">Table of n, a(n) for n = 1..10000</a>
%e A349546 a(3) = 20 is a term because A001414(20) = 2+2+5 = 9, 20+1 = 21 is divisible by 21 mod 9 = 3, and 20-1 = 19 is divisible by 19 mod 9 = 1.
%p A349546 filter:= proc(n) local s, t,r,q;
%p A349546   if isprime(n) then return false fi;
%p A349546   s:= add(t[1]*t[2],t = ifactors(n)[2]);
%p A349546   r:= (n+1) mod s;
%p A349546   q:= (n-1) mod s;
%p A349546   r<> 0 and q <> 0 and (n+1) mod r = 0 and (n-1) mod q = 0
%p A349546 end proc:
%p A349546 select(filter, [$4..2000]);
%t A349546 filter[n_] := Module[{s, t, r, q},
%t A349546    If[ PrimeQ[n], Return[False]];
%t A349546    s = Sum[t[[1]]*t[[2]], {t, FactorInteger[n]}];
%t A349546    r = Mod[n+1, s];
%t A349546    q = Mod[n-1, s];
%t A349546    r != 0 && q != 0 && Mod[n+1, r] == 0 && Mod[n-1, q ] == 0];
%t A349546 Select[Range[4, 2000], filter] (* _Jean-François Alcover_, Sep 29 2024, after Maple program *)
%Y A349546 Cf. A001414.
%Y A349546 Includes all members of A079704 except 18.
%K A349546 nonn
%O A349546 1,1
%A A349546 _J. M. Bergot_ and _Robert Israel_, Nov 21 2021