A093600 -id:A093600 - cp's OEIS frontend

A069220 Denominator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k.

1, 1, 2, 3, 12, 5, 20, 105, 280, 63, 2520, 385, 27720, 6435, 8008, 45045, 720720, 85085, 4084080, 2909907, 3695120, 1322685, 5173168, 37182145, 118982864, 128707425, 2974571600, 717084225, 80313433200, 215656441, 2329089562800

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Apr 12 2002

Seiichi Manyama, Table of n, a(n) for n = 1..2310

Cf. A017666, A002805.
Cf. A093600 (numerator of this sum).
Also denominators of row sums of A111879/A111880. For the numerators see A111881.

Table[s=0; Do[If[GCD[i, n]==1, s=s+1/i], {i, n}]; Denominator[s], {n, 1, 35}]
Table[Denominator[Total[1/Select[Range[n],GCD[n,#]==1&]]],{n,40}] (* Harvey P. Dale, Jun 07 2020 *)

for(n=1,40,print1(denominator(sum(k=1,n,if(gcd(k,n)==1,1/k))),","))

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

More terms from Jason Earls, Apr 14 2002

A290815 Numbers m such that the numerator of Sum_{k=1..m, gcd(k,m) = 1} 1/k is divisible by m^3.

Original entry on oeis.org

1, 39, 78, 155, 310, 465, 546, 793, 798, 930, 1092, 1586, 1638, 1860, 2170, 2379, 2394, 3172, 3276, 3965, 4340, 4758, 4914, 5219, 6045, 6510, 7137, 7182, 7930, 9516, 9828, 10374, 10438, 11102, 11895, 12090, 13020, 14274, 15657, 15860, 16843, 16891, 18135

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

A generalization of Wolstenholme primes (A088164) for composite number.
Leudesdorf proved in 1888 that the numerator of Sum_{k=1..n, gcd(k,n)=1} 1/k is divisible by n^2 for all (but not only) numbers n with gcd(n,6)=1, which is a generalization of Wolstenholme's theorem.
Terms that are coprime to 6: 1, 155, 793, 3965, 5219, 16843, 16891, 51305, ...
a(41) = A088164(1) = 16843.
A general conjecture: if, for some e > 0, m^e | Numerator(Sum_{k=1..m, gcd(k,m)=1} 1/k), then m^(e-1) | Numerator(Sum_{k=1..m, gcd(k,m)=1} 1/k^2). Note: in this case, the exponent e = 3. Problem: are there numbers m > 1 such that m^4 | Numerator(Sum_{k=1..m, gcd(k,m)=1} 1/k)? - Thomas Ordowski, Aug 10 2019
This general conjecture was checked up to 10^4. This problem has no solution up to 10^5. - Amiram Eldar, Aug 10 2019
It appears that all odd terms of this sequence are odd numbers m such that the numerator of Sum_{k=1..m, gcd(k,m)=1} 1/k^2 is divisible by m^2. - Thomas Ordowski, Aug 12 2019

			Sum_{k=1..39, gcd(k,39)=1} 1/k = 46855131783993/15222026943200, and 46855131783993 = 39^3 * 789884047, thus 39 is in the sequence.

G. H. Hardy and E. M. Wright, Introduction to the theory of numbers, 5th edition, Oxford, England: Clarendon Press, 1979, pp. 100-102.

David A. Corneth, Table of n, a(n) for n = 1..174 (terms <= 4*10^5; first 100 terms from Amiram Eldar)
C. Leudesdorf, Some results in the elementary theory of numbers, Proceedings of the London Mathematical Society, Vol. 20 (1888), pp. 199-212.
Eric Weisstein's World of Mathematics, Leudesdorf Theorem.
Wikipedia, Wolstenholme's theorem.

Cf. A088164, A093600.

seqQ[n_] :=  Module[{}, g[m_] := GCD[n, m] == 1; Divisible[Numerator[Plus @@ (1/Select[Range[n], g])], n^3]]; Select[Range[10^5], seqQ]

isok(n) = numerator(sum(k=1, n, if (gcd(n, k)==1, 1/k))) % n^3 == 0; \\ Michel Marcus, Aug 11 2017

upto(n) = {my(v = vector(n), d = divisors(n), res = List(), squarefreepart(n) = factorback(factorint(n)[, 1])); v[1] = 1; for(i = 2, n, v[i] = v[i-1] + 1/i; ); for(j = 1, n, fr = v[j]; d = divisors(squarefreepart(j)); for(i = 2, #d, fr += 1/d[i] * v[j/d[i]] * (-1)^omega(d[i]) ); if(numerator(fr) % j^3 == 0, listput(res, j); ) ); res } \\ David A. Corneth, Aug 23 2019

A099940 a(n) = 2(A056855(n)) /(phi(n)n), where phi() is the Euler phi function.

Original entry on oeis.org

2, 1, 1, 1, 5, 1, 84, 11, 184, 15, 193248, 23, 19056960, 833, 33740, 64035, 520105017600, 2473, 130859579289600, 203685, 963513600, 23748417, 16397141420298240000, 645119, 555804546402631680, 8527366575

Offset: 1

Leroy Quet, Nov 12 2004

nonn

Conjecture: this sequence consists completely of integers.

From Leudesdorf's theorem this is an integer sequence. - Benoit Cloitre, Nov 13 2004

			a(6) = 2*(1 + 1/5)*1*5/(6*2) = 1.

G. H. Hardy and E. M. Wright, Introduction to the theory of numbers, fifth edition, Oxford Science Publication, pp. 100-102

Eric Weisstein's World of Mathematics, Leudesdorf Theorem
Eric Weisstein's World of Mathematics, Bauers Identical Congruence

Cf. A056855, A000010.

Cf. A093600.

f[n_] := Block[{k = Select[Range[n], GCD[ #, n] == 1 &]}, 2Plus @@ (Times @@ k*Plus @@ 1/k)/EulerPhi[n]/n]; Table[ f[n], {n, 26}] (* Robert G. Wilson v, Nov 16 2004 *)

More terms from Don Reble, Nov 12 2004, who remarks that the conjecture is true for n <= 5000.

cp's OEIS Frontend

A069220 Denominator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k.

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A290815 Numbers m such that the numerator of Sum_{k=1..m, gcd(k,m) = 1} 1/k is divisible by m^3.

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A099940 a(n) = 2(A056855(n)) /(phi(n)n), where phi() is the Euler phi function.

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cp's OEIS Frontend

A069220 Denominator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k.

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A290815 Numbers m such that the numerator of Sum_{k=1..m, gcd(k,m) = 1} 1/k is divisible by m^3.

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PARI

A099940 a(n) = 2*(A056855(n)) /(phi(n)*n), where phi() is the Euler phi function.

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A099940 a(n) = 2(A056855(n)) /(phi(n)n), where phi() is the Euler phi function.