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a(n) = smallest composite squarefree number k such that (p-n) | (k+1) for all primes dividing k.
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%I #12 Jun 26 2016 10:44:03

%S 15,65,35,15,21,35,15,35,35,77,35,55,55,143,119,51,95,155,55,323,95,

%T 119,39,391,87,209,119,299,143,341,319,629,259,899,407,185,119,299,

%U 287,1517,203,799,159,155,407,1189,119,517,341,1763,1363,629,335,2491,493,3599

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

%H Paolo P. Lava, <a href="/A274444/b274444.txt">Table of n, a(n) for n = 1..250</a>

%e 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 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 with(numtheory); P:=proc(q) local d,k,n,ok,p;

%p for k from 1 to q do for n from 2 to q do

%p if not isprime(n) and issqrfree(n) then p:=ifactors(n)[2]; ok:=1;

%p for d from 1 to nops(p) do if p[d][1]=k then ok:=0; break; else

%p if not type((n+1)/(p[d][1]-k),integer) then ok:=0; break; fi; fi; od;

%p if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);

%t t = Select[Range[10^4], SquareFreeQ@ # && CompositeQ@ # &]; Table[SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k,

%t If[# == 0, False, Divisible[k + 1, #]] &[# - n] &]]], {n, 56}] (* _Michael De Vlieger_, Jun 24 2016, Version 10 *)

%Y Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274443, A274445, A274446.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, Jun 23 2016