cp's OEIS Frontend

A226112 Composite squarefree numbers n such that the ratio (n + 1/3)/(p(i) + 1/3) is an integer, where p(i) are the prime factors of n.

%I A226112 #14 Jun 04 2013 00:12:07
%S A226112 133653,1280533,193638133,514276565,1421486733,1567953933,3857178453,
%T A226112 3973200933,5411272533,7694639213,8021152533,8469827669,9820706133,
%U A226112 15832804533,18238619373,22356801133,23037766613,25136796813,27315827733,32434329685,39817016633
%N A226112 Composite squarefree numbers n such that the ratio (n + 1/3)/(p(i) + 1/3) is an integer, where p(i) are the prime factors of n.
%C A226112 Also composite squarefree numbers n such that (3*p(i)+1) | (3*n+1).
%H A226112 Giovanni Resta, <a href="/A226112/b226112.txt">Table of n, a(n) for n = 1..67</a> (terms < 2*10^12)
%e A226112 The prime factors of 133653 are 3, 13, 23 and 149. We see that (133653 + 1/3)/(3 + 1/3) = 40096, (133653 + 1/3)/(13 + 1/3) = 10024, (133653 + 1/3)/(23 + 1/3) = 5728 and (133653 + 1/3)/(149 + 1/3) = 895. Hence 133653 is in the sequence.
%e A226112 The prime factors of 1127749 are 7, 31 and 5197. We see that
%e A226112 (1127749 + 1/3)/(7 + 1/3) = 153784, (1127749 + 1/3)/(31 + 1/3) = 35992 but (1127749 + 1/3)/(5197 + 1/3) = 422906/1949. Hence 1127749 is not in the sequence.
%p A226112 with(numtheory); A226112:=proc(i, j) local c, d, n, ok, p;
%p A226112 for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
%p A226112 for d from 1 to nops(p) do if p[d][2]>1 or not type((n+j)/(p[d][1]+j),integer) then ok:=0; break; fi; od;
%p A226112 if ok=1 then print(n); fi; fi; od; end: A226112(10^9,1/3);
%Y A226112 Cf. A208728, A225702-A225720, A226020, A226111, A226113, A226114.
%K A226112 nonn,hard
%O A226112 1,1
%A A226112 _Paolo P. Lava_, May 29 2013
%E A226112 a(4)-a(21) from _Giovanni Resta_, Jun 02 2013