A226111 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.

260813, 960323, 4572113, 5991098, 18912713, 37481945, 68688458, 214337813, 1418459963, 1488523838, 1905782603, 1906387718, 2416383938, 3866147051, 6153859058, 6927221438, 10696723538, 12000312419, 24529142138, 43004079563, 43648495313, 54750300413

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..62 (terms < 2*10^12)

Example

The prime factors of 5991098 are 2, 103, 127 and 229. We see that (5991098 - 1/2)/(2 + 1/2) = 2396439, (5991098 - 1/2)/(103 + 1/2) = 57885, (5991098 - 1/2)/(127 + 1/2) = 46989 and (5991098 - 1/2)/(229 + 1/2) = 26105. Hence 5991098 is in the sequence.
The prime factors of 1123342 are 2, 11 and 51061. We see that(1123342 - 1/2)/(2 + 1/2) = 748895, (1123342 - 1/2)/(11 + 1/2) = 106985 but (1123342 - 1/2)/(51061 + 1/2) = 2246685/102121. Hence 1123342 is not in the sequence.

Maple

with(numtheory); A226111:=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: A226111(10^9, 1/2);

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226112-A226114.

Sequence in context: A184781 A146897 A249836 * A002272 A172850 A066914

Adjacent sequences: A226108 A226109 A226110 * A226112 A226113 A226114

Keyword

nonn,hard

Author

Paolo P. Lava, May 27 2013

Extensions

a(8)-a(22) from Giovanni Resta, Jun 02 2013

Status

approved