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A226112
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.
3
133653, 1280533, 193638133, 514276565, 1421486733, 1567953933, 3857178453, 3973200933, 5411272533, 7694639213, 8021152533, 8469827669, 9820706133, 15832804533, 18238619373, 22356801133, 23037766613, 25136796813, 27315827733, 32434329685, 39817016633
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..67 (terms < 2*10^12)
EXAMPLE
The prime factors of 133653 are 3, 13, 23 and 149. We see that (133653 + 1/3)/(3 + 1/3) = 40096, (133653 + 1/3)/(13 + 1/3) = 10024, (133653 + 1/3)/(23 + 1/3) = 5728 and (133653 + 1/3)/(149 + 1/3) = 895. Hence 133653 is in the sequence.
The prime factors of 1127749 are 7, 31 and 5197. We see that
(1127749 + 1/3)/(7 + 1/3) = 153784, (1127749 + 1/3)/(31 + 1/3) = 35992 but (1127749 + 1/3)/(5197 + 1/3) = 422906/1949. Hence 1127749 is not in the sequence.
MAPLE
with(numtheory); A226112:=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: A226112(10^9, 1/3);
KEYWORD
nonn,hard
AUTHOR
Paolo P. Lava, May 29 2013
EXTENSIONS
a(4)-a(21) from Giovanni Resta, Jun 02 2013
STATUS
approved