A226114 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.

1045, 1639605, 7343133, 7938133, 25615893, 282388773, 296251293, 346148733, 895445173, 1217200533, 1584568533, 2578055893, 3604398933, 4078150853, 5181367893, 5621460973, 7591692693, 8199401613, 9393224533, 9489314501, 12671984033, 12723857813, 14057815893

Offset: 1

Author

Paolo P. Lava, May 27 2013

Keywords

• nonn
• hard

Comments

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

Examples

			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.
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.
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.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..65 (terms < 2*10^12)

Crossrefs

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

Programs

Maple

with(numtheory); A226114:=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: A226114(10^9,1/3);

Extensions

a(6)-a(23) from Giovanni Resta, Jun 02 2013