A176802 a(n) = the smallest natural numbers m such that product of harmonic mean of the divisors of n and harmonic mean of the divisors of m are integers.

1, 3, 2, 7, 28, 1, 4, 420, 182, 27, 270, 14, 126, 4, 6, 31, 1638, 91, 980, 7, 32, 84, 30240, 15, 248, 63, 10, 1, 8190, 3, 16, 21, 672, 819, 4, 60515, 117800, 420, 840, 84, 55860, 4, 332640, 42, 182, 1638, 30240, 62, 380, 744, 270, 4655, 167400, 5, 54, 60, 980

Offset: 1

Author

Jaroslav Krizek, Apr 26 2010

Keywords

• nonn

Comments

Harmonic mean of the divisors of number n is rational number b(n) = n*A000005(n) / A000203(n) = A099377(n) / A099378(n).

a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A001599: a(A001599(n)) = 1. a(n) = 1 iff A099378(n) = 1.

Examples

			For n = 4; b(4) = 12/7, a(4) = 7 because b(7) = 7/4; 12/7 * 7/4 = 3 (integer).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..250

Crossrefs

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

Programs

Mathematica

h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; a[n_] := Module[{hn = h[n], k = 1}, While[! IntegerQ[hn * h[k]], k++]; k]; Array[a, 35] (* Amiram Eldar, Mar 22 2024 *)

PARI

h(n) = {my(f = factor(n)); numdiv(f)/sigma(f, -1);}
a(n) = {my(hn = h(n), k = 1); while(denominator(hn * h(k)) > 1, k++); k;} \\ Amiram Eldar, Mar 22 2024

Extensions

Data corrected and extended by Amiram Eldar, Mar 22 2024