A230587 Number n such that the sum of its proper evil divisors (A001969) equals n.

18, 476, 1484, 1988, 2324, 3164, 4172, 4564, 5516, 7196, 7364, 7532, 8036, 8876, 9716, 9772, 10052, 10444, 10892, 11956, 12572, 13076, 13412, 14084, 16604, 16772, 18004, 19866, 20692, 21328, 21364, 21644, 22316, 22988, 23492, 23884, 23996, 24164, 24668, 24836

Offset: 1

Author

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

Keywords

• nonn
• base

Comments

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.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• V. Shevelev, A question concerning perfect numbers

Crossrefs

Cf. A000396, A005101, A001969, A000069.

Programs

Mathematica

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 && dCharles R Greathouse IV, Oct 24 2013