A177050 Ceiling(n/2)-perfect numbers.

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

Offset

1,1

Comments

All powers of 2 except for 1 are terms of the sequence. All numbers of the form 2^(2^k-1)*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 ceiling(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

Amiram Eldar, Table of n, a(n) for n = 1..47

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