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A274443 Least composite squarefree number k such that (p-n) | (k-1) for all primes p dividing n.

%I A274443 #11 Jun 26 2016 10:38:19
%S A274443 561,21,85,15,21,35,33,21,65,91,57,91,133,55,161,91,57,133,33,253,65,
%T A274443 91,145,115,217,451,161,703,253,551,561,253,481,217,129,451,301,1081,
%U A274443 161,1189,145,989,217,235,481,703,649,329,265,1081,1121,1219,145,1037,721
%N A274443 Least composite squarefree number k such that (p-n) | (k-1) for all primes p dividing n.
%H A274443 Paolo P. Lava, <a href="/A274443/b274443.txt">Table of n, a(n) for n = 1..250</a>
%e A274443 Prime factors of 561 are 3, 11 and 17: (561 - 1) / (3 - 1) = 560 / 2 = 280, (561 - 1) / (11 - 1) = 560 / 10 = 56 and (561 - 1) / (17 - 1) = 560 / 16 = 35.
%e A274443 Prime factors of 21 are 3 and 7: (21 - 1) / (3 - 2) = 20 / 1 = 20, (21 - 1) / (7 - 2) = 20 / 5 = 4.
%p A274443 with(numtheory); P:=proc(q) local d,k,n,ok,p;
%p A274443 for k from 1 to q do for n from 2 to q do
%p A274443 if not isprime(n) and issqrfree(n) then p:=ifactors(n)[2]; ok:=1;
%p A274443 for d from 1 to nops(p) do if p[d][1]=k then ok:=0; break; else
%p A274443 if not type((n-1)/(p[d][1]-k),integer) then ok:=0; break; fi; fi; od;
%p A274443 if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);
%t A274443 t = Select[Range@2000, SquareFreeQ@ # && CompositeQ@ # &]; Table[SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k, If[# == 0, False, Divisible[k - 1, #]] &[# - n] &]]], {n, 55}] (* _Michael De Vlieger_, Jun 24 2016, Version 10 *)
%Y A274443 Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274444, A274445, A274446.
%K A274443 nonn,easy
%O A274443 1,1
%A A274443 _Paolo P. Lava_, Jun 23 2016