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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.
10
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
KEYWORD
nonn,hard
AUTHOR
Paolo P. Lava, May 23 2013
EXTENSIONS
a(9)-a(23) from Giovanni Resta, Jun 02 2013
STATUS
approved