A348827 Nonsquare numbers k such that the denominator of the harmonic mean of the divisors of k is larger than 2*k.

320000, 941192, 1229312, 3001250, 5120000, 8000000, 14172488, 14623232, 15059072, 19668992, 35701250, 38614472, 42762752, 60236288, 66724352, 75031250, 121726800, 128000000, 143278592, 147061250, 168480000, 222814800, 226759808, 233971712, 257875200, 319813200

Offset: 1

Author

Amiram Eldar, Nov 01 2021

Keywords

• nonn

Comments

The smallest term that is not twice a square is a(17) = 121726800 = 13 * 3060^2.

Examples

			320000 = 2 * 400^2 is a term since it is not a square, the harmonic mean of the divisors of 320000 is 16000000/798963 and 798963 > 2*320000 = 640000.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A001105, A099377, A099378.

Intersection of A000037 and A348826.

Subsequence of A005101 and A348825.

Programs

Mathematica

q[n_] := !IntegerQ @ Sqrt[n] && Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]] > 2*n; Select[Range[8000000], q]

PARI

isok(k) = if (!issquare(k), my(d=divisors(k)); (denominator(#d/sum(i=1, #d, 1/d[i])) > 2*k)); \\ Michel Marcus, Nov 01 2021

PARI

list(lim)=my(v=List()); forfactored(n=320000, lim\1, if(gcd(n[2][, 2])%2 && denominator(sigma(n, 0)/sigma(n, -1))>2*n[1], listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 01 2021