A176802 - OEIS

%I #12 Mar 22 2024 06:48:34

%S 1,3,2,7,28,1,4,420,182,27,270,14,126,4,6,31,1638,91,980,7,32,84,

%T 30240,15,248,63,10,1,8190,3,16,21,672,819,4,60515,117800,420,840,84,

%U 55860,4,332640,42,182,1638,30240,62,380,744,270,4655,167400,5,54,60,980

%N 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.

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

%C 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.

%H Amiram Eldar, <a href="/A176802/b176802.txt">Table of n, a(n) for n = 1..250</a>

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

%t 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 *)

%o (PARI) h(n) = {my(f = factor(n)); numdiv(f)/sigma(f, -1);}

%o a(n) = {my(hn = h(n), k = 1); while(denominator(hn * h(k)) > 1, k++); k;} \\ _Amiram Eldar_, Mar 22 2024

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

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Apr 26 2010

%E Data corrected and extended by _Amiram Eldar_, Mar 22 2024