login
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.
11

%I #19 Jun 04 2013 00:08:08

%S 1045,1639605,7343133,7938133,25615893,282388773,296251293,346148733,

%T 895445173,1217200533,1584568533,2578055893,3604398933,4078150853,

%U 5181367893,5621460973,7591692693,8199401613,9393224533,9489314501,12671984033,12723857813,14057815893

%N 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 Also composite squarefree numbers n such that (3*p(i) - 1) | (3*n + 1).

%H Giovanni Resta, <a href="/A226114/b226114.txt">Table of n, a(n) for n = 1..65</a> (terms < 2*10^12)

%e The prime factors of 1045 are 5, 11 and 19. We see that (1045 + 1/3)/(5 - 1/3) = 224, (1045 + 1/3)/(11 - 1/3) = 98 and (1045 + 1/3)/(19 - 1/3) = 56. Hence 1045 is in the sequence.

%e The prime factors of 1639605 are 3, 5, 11, 19 and 523. We see that (1639605 + 1/3)/(3 - 1/3) = 614852, (1639605 + 1/3)/(5 - 1/3) = 351344, (1639605 + 1/3)/(11 - 1/3) = 153713, (1639605 + 1/3)/(19 - 1/3) = 87836 and (1639605 + 1/3)/(523 - 1/3) = 3137. Hence 1639605 is in the sequence.

%e The prime factors of 1117965 are 3, 5 and 74531. We see that (1117965 + 1/3)/(3 - 1/3) = 419237, (1117965 + 1/3)/(5 - 1/3) = 239564 but (1117965 + 1/3)/(74531 - 1/3) = 419237/27949. Hence 1117965 is not in the sequence.

%p with(numtheory); A226114:=proc(i, j) local c, d, n, ok, p;

%p for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;

%p 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 if ok=1 then print(n); fi; fi; od; end: A226114(10^9,1/3);

%Y Cf. A208728, A225702-A225720, A226111-A226114.

%K nonn,hard

%O 1,1

%A _Paolo P. Lava_, May 27 2013

%E a(6)-a(23) from _Giovanni Resta_, Jun 02 2013