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A177050 Ceiling(n/2)-perfect numbers. 4

%I #25 Nov 17 2019 01:42:46

%S 2,4,8,10,16,32,64,110,128,136,256,512,884,1024,2048,4096,8192,16384,

%T 18632,32768,32896,65536,70564,100804,116624,131072,262144,391612,

%U 449295,524288,1048576,2097152,4194304,8388608,15370304,16777216,33554432,67108864,73995392

%N Ceiling(n/2)-perfect numbers.

%C 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.

%H Amiram Eldar, <a href="/A177050/b177050.txt">Table of n, a(n) for n = 1..47</a>

%o (Sage) is_A177050 = lambda n: sum(ceil(d/2) for d in divisors(n)) == 2*ceil(n/2) # _D. S. McNeil_, Dec 10 2010

%o (PARI) isok(n) = sumdiv(n, d, (d<n)*ceil(d/2)) == ceil(n/2); \\ _Michel Marcus_, Feb 08 2016

%Y Cf. A000396, A175522, A175807, A175853.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Dec 09 2010

%E a(31)-a(39) from _Michel Marcus_, Feb 08 2016

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