A090395 - OEIS

%I #28 Jun 20 2022 14:45:19

%S 1,1,3,4,5,3,7,2,3,5,11,2,13,7,15,16,17,3,19,10,21,11,23,3,25,13,27,

%T 14,29,15,31,16,33,17,35,4,37,19,39,5,41,21,43,22,15,23,47,24,49,25,

%U 51,26,53,27,55,7,57,29,59,5,61,31,21,64,65,33,67,34,69,35,71,6,73,37,25,38

%N Denominator of d(n)/n, where d(n) is the number of divisors of n (A000005).

%C The first occurrence of k (if it exists) is studied in A091895.

%C Sequence A353011 gives indices of "late birds": n such that a(k) > a(n) for all k > n. - _M. F. Hasler_, Apr 15 2022

%H Antti Karttunen, <a href="/A090395/b090395.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = n/g with g = A009191(n) = gcd(A000005(n), n). This explains the "rays" in the graph, e.g., g = 1 for odd squarefree n, g = 2 for even semiprimes n = 2p > 4 and n = 4p, p > 3. - _M. F. Hasler_, Apr 15 2022

%e a(6) = 3 because the number of divisors of 6 is 4 and 4 divided by 6 equals 2/3, which has 3 as its denominator.

%p with(numtheory): seq(denom(tau(n)/n), n=1..75) ; # _Zerinvary Lajos_, Jun 04 2008

%t Table[ Denominator[ DivisorSigma[0, n]/n], {n, 1, 80}] (* _Robert G. Wilson v_, Feb 04 2004 *)

%o (PARI) A090395(n) = denominator(numdiv(n)/n); \\ _Antti Karttunen_, Sep 25 2018

%o (Python)

%o from math import gcd

%o from sympy import divisor_count

%o def A090395(n): return n//gcd(n,divisor_count(n)) # _Chai Wah Wu_, Jun 20 2022

%Y Cf. A000005, A090387 (numerators), A091896 (numbers not in this sequence), A353011 (indices of terms such that all subsequent terms are larger).

%K easy,frac,nonn

%O 1,3

%A Ivan_E_Mayle(AT)a_provider.com, Jan 31 2004

%E More terms from _Robert G. Wilson v_, Feb 04 2004