A072158 Numerator of Sum_{k=1..n} phi(k)/k^3.

1, 9, 259, 1063, 136331, 15259, 5305837, 21351973, 1740485813, 1745820149, 2337022458319, 2341131255319, 5164765371583843, 5173292359195843, 5182536034853059, 20760610355567611, 102246457919648504843, 3789825999242633809, 26045507479622115279931, 26064975970269506857723

Offset: 1

Author

N. J. A. Sloane, Jun 28 2002

Keywords

• nonn
• frac
• easy

Examples

			1, 9/8, 259/216, 1063/864, 136331/108000, 15259/12000, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..770

Crossrefs

Cf. A000010, A072159, A306633.

Programs

GAP

List([1..25], n-> NumeratorRat( Sum([1..n], k-> Phi(k)/k^3) ) ); # G. C. Greubel, Aug 26 2019

Magma

[Numerator( &+[EulerPhi(k)/k^3: k in [1..n]] ): n in [1..25]]; // G. C. Greubel, Aug 26 2019

Maple

with(numtheory); seq(numer(add(phi(k)/k^3, k = 1..n)), n = 1..25); # G. C. Greubel, Aug 26 2019

Mathematica

Numerator[Table[Sum[EulerPhi[k]/k^3,{k,n}],{n,20}]] (* Harvey P. Dale, May 27 2012 *)
Numerator[Accumulate[Table[EulerPhi[k]/k^3, {k, 1, 30}]]] (* Amiram Eldar, Dec 28 2024 *)

PARI

a(n) = numerator( sum(k=1, n, eulerphi(k)/k^3)); \\ G. C. Greubel, Aug 26 2019

Sage

[numerator( sum(euler_phi(k)/k^3 for k in (1..n)) ) for n in (1..25)] # G. C. Greubel, Aug 26 2019

Formula

Limit_{n->oo} a(n)/A072159(n) = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Dec 28 2024