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A362804 Numbers k such that the set of divisors {d | k, BitOr(k, d) = k} has an integer harmonic mean.

%I A362804 #9 May 04 2023 14:57:57
%S A362804 1,2,4,6,8,12,16,24,28,30,32,45,48,56,60,64,90,96,112,120,128,180,192,
%T A362804 224,240,256,360,384,448,480,496,512,720,768,896,960,992,1024,1440,
%U A362804 1536,1792,1920,1984,2048,2880,3072,3584,3840,3968,4096,5760,6144,7168,7680
%N A362804 Numbers k such that the set of divisors {d | k, BitOr(k, d) = k} has an integer harmonic mean.
%C A362804 Equivalently, the set of divisors can be defined by {d | k, BitAnd(k, d) = d}.
%C A362804 Analogous to harmonic (or Ore) numbers (A001599) where the divisors d of k are restricted by BitOr(k, d) = k or BitAnd(k, d) = d.
%C A362804 If k is a term then so is 2*k. The primitive terms are in A362805. Thus, this sequence includes all the powers of 2 (A000079), all the numbers of the form 3*2^m and 15*2^m for m >= 1, and all the numbers of the form 7*2^m for m >= 2.
%C A362804 All the even perfect numbers (A000396) are terms: if k = 2^(p-1)*(2^p-1) is a perfect number (where p is a Mersenne exponent, A000043), then the only divisors of k such that BitOr(k, d) = k are 2^(p-1) and k itself, and the harmonic mean of 2^(p-1) and 2^(p-1)*(2^p-1) is 2^p - 1.
%C A362804 Are 1 and 45 the only odd terms in this sequence?
%H A362804 Amiram Eldar, <a href="/A362804/b362804.txt">Table of n, a(n) for n = 1..406</a>
%t A362804 q[n_] := IntegerQ[HarmonicMean[Select[Divisors[n], BitAnd[n, #] == # &]]]; Select[Range[10^4], q]
%o A362804 (PARI) div(n) = select(x->(bitor(x, n) == n), divisors(n));
%o A362804 is(n) = {my(d = div(n)); denominator(#d/sum(i = 1, #d, 1/d[i])) == 1;}
%Y A362804 Cf. A000043, A000396, A246600, A246601.
%Y A362804 Subsequences: A000079, A007283 \ {3}, A005009 \ {7, 14}, A110286 \ {15}, A362805.
%Y A362804 Similar sequences: A001599, A006086, A063947, A286325, A319745.
%K A362804 nonn,base
%O A362804 1,2
%A A362804 _Amiram Eldar_, May 04 2023