A348866 Composite numbers k such that A099378(k) | (A099377(k) + 1).

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

Offset: 1

Author

Amiram Eldar, Nov 02 2021

Keywords

• nonn

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).

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000203, A001599, A003627, A099377, A099378, A348865.

Programs

Mathematica

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