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A274444 a(n) = smallest composite squarefree number k such that (p-n) | (k+1) for all primes dividing k.

%I A274444 #12 Jun 26 2016 10:44:03
%S A274444 15,65,35,15,21,35,15,35,35,77,35,55,55,143,119,51,95,155,55,323,95,
%T A274444 119,39,391,87,209,119,299,143,341,319,629,259,899,407,185,119,299,
%U A274444 287,1517,203,799,159,155,407,1189,119,517,341,1763,1363,629,335,2491,493,3599
%N A274444 a(n) = smallest composite squarefree number k such that (p-n) | (k+1) for all primes dividing k.
%H A274444 Paolo P. Lava, <a href="/A274444/b274444.txt">Table of n, a(n) for n = 1..250</a>
%e A274444 a(1) = 15: Prime factors of 15 are 3 and 5: (15 + 1) / (3 - 1) = 16 / 2 = 8 and (15 + 1) / (5 - 1) = 16 / 4 = 4.
%e A274444 a(2) = 6: Prime factors of 65 are 5 and 13: (65 + 1) / (5 - 2) = 66 / 3 = 22 and (65 + 1) / (13 - 2) = 66 / 11 = 6.
%p A274444 with(numtheory); P:=proc(q) local d,k,n,ok,p;
%p A274444 for k from 1 to q do for n from 2 to q do
%p A274444 if not isprime(n) and issqrfree(n) then p:=ifactors(n)[2]; ok:=1;
%p A274444 for d from 1 to nops(p) do if p[d][1]=k then ok:=0; break; else
%p A274444 if not type((n+1)/(p[d][1]-k),integer) then ok:=0; break; fi; fi; od;
%p A274444 if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);
%t A274444 t = Select[Range[10^4], SquareFreeQ@ # && CompositeQ@ # &]; Table[SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k,
%t A274444 If[# == 0, False, Divisible[k + 1, #]] &[# - n] &]]], {n, 56}] (* _Michael De Vlieger_, Jun 24 2016, Version 10 *)
%Y A274444 Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274443, A274445, A274446.
%K A274444 nonn,easy
%O A274444 1,1
%A A274444 _Paolo P. Lava_, Jun 23 2016