A346400 Composite numbers k such that the numerator of the harmonic mean of the divisors of k is equal to k.

20, 21, 22, 27, 35, 38, 39, 45, 49, 55, 56, 57, 65, 68, 77, 85, 86, 93, 99, 110, 111, 115, 116, 118, 119, 125, 129, 133, 134, 143, 147, 150, 155, 161, 164, 166, 169, 183, 184, 185, 187, 189, 201, 203, 205, 207, 209, 212, 214, 215, 217, 219, 221, 235, 237, 245

Offset

1,1

Comments

Composite numbers k such that A099377(k) = k.

Since the harmonic mean of the divisors of an odd prime p is p/((p+1)/2), its numerator is equal to p. Therefore, this sequence is restricted to composite numbers.

This sequence is infinite. For example, if p is a prime of the form 8*k+3 (A007520) with k>1, then 2*p is a term.

Links

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

Example

20 is a term since the harmonic mean of the divisors of 20 is 20/7.

Mathematica

q[n_] := CompositeQ[n] && Numerator[DivisorSigma[0, n]/DivisorSigma[-1, n]] == n; Select[Range[250], q]

Prog

(PARI) isok(k) = my(d=divisors(k)); (#d>2) && (numerator(#d/sum(i=1, #d, 1/d[i])) == k); \\ Michel Marcus, Nov 01 2021
(PARI) list(lim)=my(v=List()); forfactored(n=20, lim\1, if(vecsum(n[2][, 2])>1 && numerator(sigma(n, 0)/sigma(n, -1))==n[1], listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 01 2021

Crossrefs

Intersection of A002808 and A250094.

Cf. A099377, A099378.

Sequence in context: A270045 A133895 A138602 * A347298 A326678 A347469

Adjacent sequences: A346397 A346398 A346399 * A346401 A346402 A346403

Keyword

nonn

Author

Amiram Eldar, Nov 01 2021

Status

approved