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.

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);

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111, A226113, A226114.

Sequence in context: A061732 A023046 A034213 * A244211 A234206 A133528

Adjacent sequences: A226109 A226110 A226111 * A226113 A226114 A226115

Keyword

nonn,hard

Author

Paolo P. Lava, May 29 2013

Extensions

a(4)-a(21) from Giovanni Resta, Jun 02 2013

Status

approved