A230587 - OEIS

%I #22 Jan 02 2023 12:30:49

%S 18,476,1484,1988,2324,3164,4172,4564,5516,7196,7364,7532,8036,8876,

%T 9716,9772,10052,10444,10892,11956,12572,13076,13412,14084,16604,

%U 16772,18004,19866,20692,21328,21364,21644,22316,22988,23492,23884,23996,24164,24668,24836

%N Number n such that the sum of its proper evil divisors (A001969) equals n.

%C Sequence could be called the "evil-perfect numbers".

%C By the Euclid-Euler theorem, an even number n is perfect (A000396) if and only if n=2^(k-1)*(2^k-1), where 2^k-1 is prime. From this it follows that all even perfect numbers more than 6 have only odious divisors (A000069). In contrast to them, this sequence lists those abundant numbers n (A005101), all proper evil divisors of which sum to n.

%C It is asked, are there non-perfect numbers n, all proper odious divisors of which sum to n? The first two such numbers were found by Giovanni Resta, see A212302.

%H Amiram Eldar, <a href="/A230587/b230587.txt">Table of n, a(n) for n = 1..10000</a>

%H V. Shevelev, <a href="http://list.seqfan.eu/oldermail/seqfan/2013-October/011795.html">A question concerning perfect numbers</a>

%e 18 is in the sequence since its proper divisors are {1, 2, 3, 6, 9}, and their subset that is in A001969 is {3, 6, 9} whose sum is 18.

%t aQ[n_] := DivisorSum[n, # &, # < n && EvenQ[DigitCount[#, 2][[1]]] &] == n; Select[Range[25000], aQ] (* _Amiram Eldar_, Jun 21 2019 *)

%o (PARI) is(n)=sumdiv(n,d,if(hammingweight(d)%2==0 && d<n,d))==n \\ _Charles R Greathouse IV_, Oct 24 2013

%Y Cf. A000396, A005101, A001969, A000069.

%K nonn,base

%O 1,1

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Oct 24 2013