A093600 - OEIS

%I #29 Sep 08 2022 08:45:13

%S 1,1,3,4,25,6,49,176,621,100,7381,552,86021,11662,18075,91072,2436559,

%T 133542,14274301,5431600,9484587,2764366,19093197,61931424,399698125,

%U 281538452,8770427199,1513702904,315404588903,323507400,9304682830147

%N Numerator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k.

%C The divisibility properties of this sequence are given by Leudesdorf's theorem.

%C Problem: are there numbers n > 1 such that n^4 | a(n)? Let b(n) be the numerator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k^2. Conjecture: if, for some e > 0, n^e | a(n), then n^(e-1) | b(n). It appears that, for any odd number n, n^e | a(n) if and only if n^(e-1) | b(n). - _Thomas Ordowski_, Aug 12 2019

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1971, page 100. [3rd. ed., Theorem 128, page 101]

%H Seiichi Manyama, <a href="/A093600/b093600.txt">Table of n, a(n) for n = 1..2310</a>

%H Emre Alkan, <a href="http://www.jstor.org/stable/2975168">Variations on Wolstenholme's Theorem</a>, Amer. Math. Monthly, Vol. 101, No. 10 (Dec. 1994), 1001-1004.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeudesdorfTheorem.html">Leudesdorf Theorem</a>

%F G.f. A(x) (for fractions) satisfies: A(x) = -log(1 - x)/(1 - x) - Sum_{k>=2} A(x^k)/k. - _Ilya Gutkovskiy_, Mar 31 2020

%t Table[s=0; Do[If[GCD[i, n]==1, s=s+1/i], {i, n}]; Numerator[s], {n, 1, 35}]

%o (PARI) for (n=1, 40, print1(numerator(sum(k=1, n, if (gcd(k, n)==1, 1/k))), ", ")) \\ _Seiichi Manyama_, Aug 11 2017

%o (Magma) [Numerator(&+[1/k:k in [1..n]|Gcd(k,n) eq 1]):n in [1..31]]; // _Marius A. Burtea_, Aug 14 2019

%Y Cf. A069220 (denominator of this sum), A001008 (numerator of the n-th harmonic number).

%K nonn,frac

%O 1,3

%A _T. D. Noe_, Apr 03 2004