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A093600 Numerator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

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

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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

Sequence in context: A256830 A065900 A065809 * A128778 A338425 A304210

Adjacent sequences:  A093597 A093598 A093599 * A093601 A093602 A093603

KEYWORD

nonn,frac

AUTHOR

T. D. Noe, Apr 03 2004

STATUS

approved

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Last modified August 4 16:18 EDT 2021. Contains 346447 sequences. (Running on oeis4.)