A120263 - OEIS

%I #17 Jan 26 2023 10:05:47

%S 1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,3,1,5,3,1,1,1,1,1,1,1,1,1,1,1,11,1,

%T 1,1,1,1,1,1,1,7,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,3,1,1,3,1,1,

%U 1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,25,1,1,1,1

%N Ratio of the numerator of n*HarmonicNumber[n] to the numerator of HarmonicNumber[n]: A096617(n)/A001008(n).

%C a(n) is not equal to 1 when n belongs to A074791 - numbers n such that n does not divide the denominator of the n-th harmonic number.

%C a(n) is almost always equal to 1 except for n=6,18,20,21,33,42,54,.. when a(n) seems to be equal to a prime divisor of n.

%C a(n) could be equal to a squared prime divisor of n as for n=100,294,500,847,..

%H G. C. Greubel, <a href="/A120263/b120263.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A096617(n)/A001008(n) = numerator[n*Sum[1/i,{i,1,n}]] / numerator[Sum[1/i,{i,1,n}]].

%F a(n) = n / gcd(denominator(H(n)),n), where H(n) = sum(1/k, k=1..n). [_Gary Detlefs_, Sep 05 2011]

%F a(n) = A096617(n)*A110566(n)/A025529(n). [_Arkadiusz Wesolowski_, Mar 29 2012]

%t Numerator[Table[n*Sum[1/i,{i,1,n}],{n,1,500}]]/Numerator[Table[Sum[1/i,{i,1,n}],{n,1,500}]]

%o (PARI) {h(n) = sum(k=1,n,1/k)};

%o for(n=1,100, print1(numerator(n*h(n))/numerator(h(n)), ", ")) \\ _G. C. Greubel_, Sep 01 2018

%o (Magma) [Numerator(n*HarmonicNumber(n))/Numerator(HarmonicNumber(n)): n in [1..100]]; // _G. C. Greubel_, Sep 01 2018

%Y Cf. A096617, A001008, A074791.

%K frac,nonn

%O 1,6

%A _Alexander Adamchuk_, Jun 26 2006