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

1,2

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.

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A248054 A329421 A100985 * A363400 A230710 A265009

Adjacent sequences: A176799 A176800 A176801 * A176803 A176804 A176805

Keyword

nonn

Author

Jaroslav Krizek, Apr 26 2010

Extensions

Data corrected and extended by Amiram Eldar, Mar 22 2024

Status

approved