

A348866


Composite numbers k such that A099378(k)  (A099377(k) + 1).


2



6, 15, 20, 28, 33, 35, 42, 51, 66, 69, 70, 84, 87, 114, 117, 123, 135, 138, 140, 141, 153, 159, 177, 186, 204, 207, 210, 213, 249, 258, 267, 270, 273, 276, 282, 285, 297, 303, 308, 321, 339, 348, 354, 357, 372, 393, 399, 402, 411, 420, 426, 432, 435, 447, 464
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OFFSET

1,1


COMMENTS

A disjoint union of the harmonic numbers (A001599) and the composite numbers whose harmonic mean of divisors is of the form m  1/k, where m and k are integers.
If p is an odd prime, then the harmonic mean of its divisors is p*tau(p)/sigma(p) = p*A000005(p)/A000203(p) = p/((p+1)/2), so A099378(p)  (A099377(p) + 1). Therefore, this sequence is restricted to composite numbers.
This sequence is infinite. For example, it includes all the semiprimes of the form 3*p, where p == 2 (mod 3).


LINKS



EXAMPLE

15 is a term since it is composite, the harmonic mean of divisors of 15 is 5/2 and 2  (5+1).


MATHEMATICA

h[n_] := DivisorSigma[0, n]/DivisorSigma[1, n]; q[n_] := Divisible[Numerator[(h1 = h[n])] + 1, Denominator[h1]]; Select[Range[1000], CompositeQ[#] && q[#] &]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



