1,1

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

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.

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

V. Shevelev, A question concerning perfect numbers

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.

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

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

Cf. A000396, A005101, A001969, A000069.

Sequence in context: A053115 A084273 A386771 * A281161 A027405 A382404

Adjacent sequences: A230584 A230585 A230586 * A230588 A230589 A230590

nonn,base

Vladimir Shevelev and Peter J. C. Moses, Oct 24 2013

approved