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%I #20 Jul 16 2021 21:35:08
%S 28,496,8128,415800,2096128,33550336,8589869056
%N Numbers k whose sum of proper odious divisors (A000069) equals k.
%C Sequence could be called the "odious-perfect numbers".
%C 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.
%t 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 *)
%o (PARI) is(n)=sumdiv(n,d,if(hammingweight(d)%2 && d<n,d))==n \\ _Charles R Greathouse IV_, Oct 24 2013
%Y Cf. A230587, A000396, A005101, A000069, A001969.
%K nonn,base,more
%O 1,1
%A _Vladimir Shevelev_, _Peter J. C. Moses_, and _Giovanni Resta_, Oct 24 2013
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