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)).

4, 8, 20, 32, 50, 55, 64, 77, 80, 98, 110, 115, 125, 128, 152, 170, 216, 242, 243, 256, 275, 290, 329, 338, 341, 343, 364, 371, 416, 506, 511, 512, 544, 551, 578, 583, 611, 638, 663, 722, 729, 731, 741, 851, 870, 920, 987, 1024, 1025, 1054, 1058, 1079, 1144, 1219, 1243, 1298, 1325, 1331, 1421

Offset: 1

Author

J. M. Bergot and Robert Israel, Nov 21 2021

Keywords

• nonn

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414.

Includes all members of A079704 except 18.

Programs

Maple

filter:= proc(n) local s, t,r,q;
  if isprime(n) then return false fi;
  s:= add(t[1]*t[2],t = ifactors(n)[2]);
  r:= (n+1) mod s;
  q:= (n-1) mod s;
  r<> 0 and q <> 0 and (n+1) mod r = 0 and (n-1) mod q = 0
end proc:
select(filter, [$4..2000]);

Mathematica

filter[n_] := Module[{s, t, r, q},
   If[ PrimeQ[n], Return[False]];
   s = Sum[t[[1]]*t[[2]], {t, FactorInteger[n]}];
   r = Mod[n+1, s];
   q = Mod[n-1, s];
   r != 0 && q != 0 && Mod[n+1, r] == 0 && Mod[n-1, q ] == 0];
Select[Range[4, 2000], filter] (* Jean-François Alcover, Sep 29 2024, after Maple program *)