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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.
2

%I #22 Jun 04 2013 03:10:38

%S 773227,13596427,26567147,140247467,525558107,1390082027,1847486667,

%T 2514565387,3699765755,4060724267,4520219947,6185512667,6480142667,

%U 8328046827,9951353867,10268992067,11720901387,14149448387,14913513067,21926400427,22367433387,24260249387

%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="/A226113/b226113.txt">Table of n, a(n) for n = 1..77</a> (terms < 2*10^12)

%e The prime factors of 773227 are 7, 13, 29 and 293. We see that (773227 - 1/3)/(7 + 1/3) = 231968, (773227 - 1/3)/(13 + 1/3) = 57992, (773227 - 1/3)/(29 + 1/3) = 26360 and (773227 - 1/3)/(293 + 1/3) = 2636. Hence 773227 is in the sequence.

%e The prime factors of 1128387 are 3, 13 and 28933. We see that

%e (1128387 - 1/3)/(3 + 1/3) = 338516, (1128387 - 1/3)/(13 + 1/3) = 84629 but (1128387 - 1/3)/(28933 + 1/3) = 84629/2170. Hence 1128387 is not in the sequence.

%p with(numtheory); A226113:=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: A226113(10^9,1/3);

%Y Cf. A208728, A225702-A225720, A226020, A226111, A226112, A226114.

%K nonn,hard

%O 1,1

%A _Paolo P. Lava_, May 29 2013

%E a(5)-a(22) from _Giovanni Resta_, Jun 02 2013