

A177050


Ceil(n/2)perfect numbers


3



2, 4, 8, 10, 16, 32, 64, 110, 128, 136, 256, 512, 884, 1024, 2048, 4096, 8192, 16384, 18632, 32768, 32896, 65536, 70564, 100804, 116624, 131072, 262144, 391612, 449295, 524288, 1048576, 2097152, 4194304, 8388608, 15370304, 16777216, 33554432, 67108864, 73995392
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

All powers of 2 except for 1 are terms of the sequence. All numbers of the form 2^(2^k1)*p, where p=2^(2^k)+1 is a Fermat prime (k >= 1) are in the sequence. Thus numbers 136, 32896, 2147516416 are in the sequence. It is interesting that in this construction Fermat primes play the same role that Mersenne primes in construction of usual even perfect numbers. Unfortunately, the conversion for even ceil(n/2)perfect numbers is false: the first counterexample, found by D. S. McNeil, is 110=2*5*11. Besides, the first odd term, found by D. S. McNeil, is 449295=3 * 5 * 7 * 11 * 389.


LINKS

Table of n, a(n) for n=1..39.


PROG

(Sage) is_A177050 = lambda n: sum(ceil(d/2) for d in divisors(n)) == 2*ceil(n/2) # D. S. McNeil, Dec 10 2010
(PARI) isok(n) = sumdiv(n, d, (d<n)*ceil(d/2)) == ceil(n/2); \\ Michel Marcus, Feb 08 2016


CROSSREFS

Cf. A000396, A175522, A175807, A175853.
Sequence in context: A097210 A097214 A045579 * A276772 A173063 A066363
Adjacent sequences: A177047 A177048 A177049 * A177051 A177052 A177053


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Dec 09 2010


EXTENSIONS

a(31)a(39) from Michel Marcus, Feb 08 2016


STATUS

approved



