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

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

Offset

1,1

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.

Links

Table of n, a(n) for n=1..7.

Mathematica

okQ[n_] := DivisorSum[n, If[OddQ[Total[IntegerDigits[#, 2]]] && #<n, #, 0] &] == n; Reap[For[n=1, n<9*10^9, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Dec 06 2015 *)

Prog

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

Crossrefs

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

Sequence in context: A076172 A004336 A079598 * A046999 A046986 A330164

Adjacent sequences: A212299 A212300 A212301 * A212303 A212304 A212305

Keyword

nonn,base,more

Author

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

Status

approved