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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.
11
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
KEYWORD
nonn,hard
AUTHOR
Paolo P. Lava, May 27 2013
EXTENSIONS
a(6)-a(23) from Giovanni Resta, Jun 02 2013
STATUS
approved