

A177050


Ceiling(n/2)perfect numbers.


4



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
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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 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



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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



