A226020 Composite squarefree numbers n such that the ratio (n + 1/2)/(p(i) + 1/2) is an integer, where p(i) are the prime factors of n.

13702, 42997, 1004062, 1684462, 38447662, 40243549, 70801087, 107728582, 409055062, 594021862, 760767262, 1045475437, 1104435202, 1471700587, 1686747562, 1920806662, 3136180162, 3469071937, 5291041297, 7239716347, 7903353667, 12738885862, 22711489762

Offset

1,1

Comments

Also composite squarefree numbers n such that (2*p(i)+1) | (2*n+1).

Links

Giovanni Resta, Table of n, a(n) for n = 1..65 (terms < 3*10^12)

Example

The prime factors of 13702 are 2, 13, 17 and 31. We see that (13702 + 1)/(2 + 1/2) = 5481, (13702 + 1/2)/(13 + 1/2) = 1015, (13702 + 1)/(17 + 1/2) = 783 and ( 13702 + 1/2)/(31 + 1/2) = 435. Hence 13702 is in the sequence.
The prime factors of 1123545 are 3, 5 and 74903. We see that
(1123545 + 1/2)/(3 + 1/2) = 321013, (1123545 + 1/2)/(5 + 1/2) = 204281 but (1123545 + 1/2)/(74903+ 1/2) = 321013/21401. Hence 1123545 is not in the sequence.

Maple

with(numtheory); A226020:=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: A226020(10^9, 1/2);

Crossrefs

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

Sequence in context: A088870 A047827 A204369 * A157669 A236052 A178486

Adjacent sequences: A226017 A226018 A226019 * A226021 A226022 A226023

Keyword

nonn,hard

Author

Paolo P. Lava, May 23 2013

Extensions

a(9)-a(23) from Giovanni Resta, Jun 02 2013

Status

approved