A071708 Numerator of Sum_{k=1..n} phi(k)/k.

1, 3, 13, 8, 52, 19, 163, 361, 1223, 1307, 16477, 5749, 83977, 88267, 280817, 147916, 2754812, 2839897, 58552633, 60492571, 63263911, 65468386, 1612469468, 549883871, 579629587, 596790577, 1864736021, 1912541636, 58587007624, 59449633388, 1939984033478

Offset: 1

Author

N. J. A. Sloane, Jun 28 2002

Keywords

• nonn
• frac

Examples

			1, 3/2, 13/6, 8/3, 52/15, 19/5, 163/35, 361/70, 1223/210, ...
		

References

• József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section I.24, page 27.
• Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Berlin, 1963.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A072155 (denominators), A000010, A059956.

Programs

GAP

List([1..35], n-> NumeratorRat( Sum([1..n], k-> Phi(k)/k) ) ); # G. C. Greubel, Aug 25 2019

Magma

[Numerator( &+[EulerPhi(k)/k: k in [1..n]] ): n in [1..35]]; // G. C. Greubel, Aug 25 2019

Maple

with(numtheory); seq(numer(add(phi(k)/k, k = 1..n)), n = 1..35); # G. C. Greubel, Aug 25 2019

Mathematica

Table[Sum[EulerPhi[k]/k, {k, n}], {n,35}]//Numerator (* G. C. Greubel, Aug 25 2019 *)

PARI

a(n) = numerator(sum(k=1,n, eulerphi(k)/k));
vector(35, n, a(n)) \\ G. C. Greubel, Aug 25 2019

Sage

[numerator( sum(euler_phi(k)/k for k in (1..n)) ) for n in (1..35)] # G. C. Greubel, Aug 25 2019

Formula

Also numerator of Sum_{i=1..n} (mu(i)/i)*floor(n/i). - Ridouane Oudra, Nov 26 2019

a(n)/A072155(n) ~ (6/Pi^2) * n + O(log(n)^(2/3)*log(log(n))^(4/3)). - Amiram Eldar, Sep 18 2022