A226113 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.

773227, 13596427, 26567147, 140247467, 525558107, 1390082027, 1847486667, 2514565387, 3699765755, 4060724267, 4520219947, 6185512667, 6480142667, 8328046827, 9951353867, 10268992067, 11720901387, 14149448387, 14913513067, 21926400427, 22367433387, 24260249387

Offset

1,1

Comments

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

Links

Giovanni Resta, Table of n, a(n) for n = 1..77 (terms < 2*10^12)

Example

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

Maple

with(numtheory); A226113:=proc(i, j) local c, d, n, ok, p;
for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
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;
if ok=1 then print(n); fi; fi; od; end: A226113(10^9, 1/3);

Crossrefs

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

Sequence in context: A237573 A232123 A232421 * A151562 A023348 A089007

Adjacent sequences: A226110 A226111 A226112 * A226114 A226115 A226116

Keyword

nonn,hard

Author

Paolo P. Lava, May 29 2013

Extensions

a(5)-a(22) from Giovanni Resta, Jun 02 2013

Status

approved