cp's OEIS Frontend

A226020 Composite squarefree numbers n such that the ratio (n + 1/2)/(p(i) + 1/2) is an integer, where p(i) are the prime factors of n.

%I A226020 #18 Jun 04 2013 00:16:00
%S A226020 13702,42997,1004062,1684462,38447662,40243549,70801087,107728582,
%T A226020 409055062,594021862,760767262,1045475437,1104435202,1471700587,
%U A226020 1686747562,1920806662,3136180162,3469071937,5291041297,7239716347,7903353667,12738885862,22711489762
%N A226020 Composite squarefree numbers n such that the ratio (n + 1/2)/(p(i) + 1/2) is an integer, where p(i) are the prime factors of n.
%C A226020 Also composite squarefree numbers n such that (2*p(i)+1) | (2*n+1).
%H A226020 Giovanni Resta, <a href="/A226020/b226020.txt">Table of n, a(n) for n = 1..65</a> (terms < 3*10^12)
%e A226020 The prime factors of 13702 are 2, 13, 17 and 31. We see that (13702 + 1)/(2 + 1/2) = 5481, (13702 + 1/2)/(13 + 1/2) = 1015, (13702 + 1)/(17 + 1/2) = 783 and ( 13702 + 1/2)/(31 + 1/2) = 435. Hence 13702 is in the sequence.
%e A226020 The prime factors of 1123545 are 3, 5 and 74903. We see that
%e A226020 (1123545 + 1/2)/(3 + 1/2) = 321013, (1123545 + 1/2)/(5 + 1/2) = 204281 but (1123545 + 1/2)/(74903+ 1/2) = 321013/21401. Hence 1123545 is not in the sequence.
%p A226020 with(numtheory); A226020:=proc(i, j) local c, d, n, ok, p;
%p A226020 for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
%p A226020 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 A226020 if ok=1 then print(n); fi; fi; od; end: A226020(10^9,1/2);
%Y A226020 Cf. A208728, A225702-A225720, A226111-A226114.
%K A226020 nonn,hard
%O A226020 1,1
%A A226020 _Paolo P. Lava_, May 23 2013
%E A226020 a(9)-a(23) from _Giovanni Resta_, Jun 02 2013