cp's OEIS Frontend

A226111 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 A226111 #20 Jun 04 2013 03:30:45
%S A226111 260813,960323,4572113,5991098,18912713,37481945,68688458,214337813,
%T A226111 1418459963,1488523838,1905782603,1906387718,2416383938,3866147051,
%U A226111 6153859058,6927221438,10696723538,12000312419,24529142138,43004079563,43648495313,54750300413
%N A226111 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 A226111 Also composite squarefree numbers n such that (2*p(i)+1) | (2*n-1).
%H A226111 Giovanni Resta, <a href="/A226111/b226111.txt">Table of n, a(n) for n = 1..62</a> (terms < 2*10^12)
%e A226111 The prime factors of 5991098 are 2, 103, 127 and 229. We see that (5991098 - 1/2)/(2 + 1/2) = 2396439, (5991098 - 1/2)/(103 + 1/2) = 57885, (5991098 - 1/2)/(127 + 1/2) = 46989 and (5991098 - 1/2)/(229 + 1/2) = 26105. Hence 5991098 is in the sequence.
%e A226111 The prime factors of 1123342 are 2, 11 and 51061. We see that(1123342 - 1/2)/(2 + 1/2) = 748895, (1123342 - 1/2)/(11 + 1/2) = 106985 but (1123342 - 1/2)/(51061 + 1/2) = 2246685/102121. Hence 1123342 is not in the sequence.
%p A226111 with(numtheory); A226111:=proc(i, j) local c, d, n, ok, p;
%p A226111 for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
%p A226111 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 A226111 if ok=1 then print(n); fi; fi; od; end: A226111(10^9,1/2);
%Y A226111 Cf. A208728, A225702-A225720, A226020, A226112-A226114.
%K A226111 nonn,hard
%O A226111 1,1
%A A226111 _Paolo P. Lava_, May 27 2013
%E A226111 a(8)-a(22) from _Giovanni Resta_, Jun 02 2013