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

Author

Paolo P. Lava, May 29 2013

Keywords

• nonn
• hard

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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);

Extensions

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