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

1,1

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..65 (terms < 2*10^12)

Example

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.

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);

Crossrefs

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

Sequence in context: A344410 A110597 A258130 * A249653 A056092 A280809

Adjacent sequences: A226111 A226112 A226113 * A226115 A226116 A226117

Keyword

nonn,hard

Author

Paolo P. Lava, May 27 2013

Extensions

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

Status

approved