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

Author

Amiram Eldar, Nov 01 2021

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Intersection of A002808 and A250094.

Cf. A099377, A099378.

Programs

Mathematica

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

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