A093600 Numerator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k.

1, 1, 3, 4, 25, 6, 49, 176, 621, 100, 7381, 552, 86021, 11662, 18075, 91072, 2436559, 133542, 14274301, 5431600, 9484587, 2764366, 19093197, 61931424, 399698125, 281538452, 8770427199, 1513702904, 315404588903, 323507400, 9304682830147

Offset: 1

Author

T. D. Noe, Apr 03 2004

Keywords

• nonn
• frac

Comments

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

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

References

• 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]

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..2310
• Emre Alkan, Variations on Wolstenholme's Theorem, Amer. Math. Monthly, Vol. 101, No. 10 (Dec. 1994), 1001-1004.
• Eric Weisstein's World of Mathematics, Leudesdorf Theorem

Crossrefs

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

Programs

Magma

[Numerator(&+[1/k:k in [1..n]|Gcd(k,n) eq 1]):n in [1..31]]; // Marius A. Burtea, Aug 14 2019

Mathematica

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

PARI

for (n=1, 40, print1(numerator(sum(k=1, n, if (gcd(k, n)==1, 1/k))), ", ")) \\ Seiichi Manyama, Aug 11 2017

Formula

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