A212302 Numbers k whose sum of proper odious divisors (A000069) equals k.

28, 496, 8128, 415800, 2096128, 33550336, 8589869056

Offset: 1

Author

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

Keywords

• nonn
• base
• more

Comments

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

By the Euclid-Euler theorem, an even number k is perfect (A000396) if and only if k = 2^(m-1)*(2^m-1), where 2^m-1 is prime. From this it follows that all even perfect numbers greater than 6 have only odious divisors (A000069). Therefore, they all are in this sequence. However, the sequence also contains non-perfect numbers. The first two such numbers are 415800 and 2096128 (and no others up to 10^11). Comparing this sequence with A230587, one can see that the terms of this sequence are much rarer. It is an interesting phenomenon.

Crossrefs

Cf. A230587, A000396, A005101, A000069, A001969.

Programs

Mathematica

okQ[n_] := DivisorSum[n, If[OddQ[Total[IntegerDigits[#, 2]]] && #Jean-François Alcover, Dec 06 2015 *)

PARI

is(n)=sumdiv(n,d,if(hammingweight(d)%2 && dCharles R Greathouse IV, Oct 24 2013