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

1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 45, 48, 56, 60, 64, 90, 96, 112, 120, 128, 180, 192, 224, 240, 256, 360, 384, 448, 480, 496, 512, 720, 768, 896, 960, 992, 1024, 1440, 1536, 1792, 1920, 1984, 2048, 2880, 3072, 3584, 3840, 3968, 4096, 5760, 6144, 7168, 7680

Offset

1,2

Comments

Equivalently, the set of divisors can be defined by {d | k, BitAnd(k, d) = d}.

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.

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.

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.

Are 1 and 45 the only odd terms in this sequence?

Links

Amiram Eldar, Table of n, a(n) for n = 1..406

Mathematica

q[n_] := IntegerQ[HarmonicMean[Select[Divisors[n], BitAnd[n, #] == # &]]]; Select[Range[10^4], q]

Prog

(PARI) div(n) = select(x->(bitor(x, n) == n), divisors(n));
is(n) = {my(d = div(n)); denominator(#d/sum(i = 1, #d, 1/d[i])) == 1; }

Crossrefs

Cf. A000043, A000396, A246600, A246601.

Subsequences: A000079, A007283 \ {3}, A005009 \ {7, 14}, A110286 \ {15}, A362805.

Similar sequences: A001599, A006086, A063947, A286325, A319745.

Sequence in context: A058629 A323508 A324850 * A095810 A025487 A333964

Adjacent sequences: A362801 A362802 A362803 * A362805 A362806 A362807

Keyword

nonn,base

Author

Amiram Eldar, May 04 2023

Status

approved