A048290 Numbers m that divide Sum_{k=1..m} phi(k).

1, 2, 5, 6, 16, 25, 36, 249, 617, 1296, 13763, 76268, 189074, 783665, 1102394, 3258466, 3808854, 7971034, 15748051, 27746990, 41846733, 153673168, 195853251, 302167272, 402296412, 732683468, 807656448, 844492262, 848152352, 1122039882, 2258200198, 2438160726

Offset: 1

Author

David J. Rusin

Keywords

• nonn
• nice

Comments

The odd terms of this sequence and A063986 are the same. - Jud McCranie, Jun 26 2005

Examples

			Euler sums are *1*, *2*, 4, 6, *10*, *12*, ..., *80*, ..., *510624*,... for n=1, 2, 3, 4, 5, 6, ..., 16, ...., 1296, ...
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..37 (terms < 10^12)
• Edward A. Bender, Oren Patashnik and Howard Rumsey, Jr., Pizza Slicing, Phi's and the Riemann Hypothesis, American Mathematical Monthly, Vol. 101 (1994), pp. 307-317.
• D. Rusin, Euler phi function

Crossrefs

Cf. A000010, A002088. See A063986 for n divides Sum_{k=1..n} k-phi(k).

Programs

Mathematica

s = 0; Do[s = s + EulerPhi[n]; If[IntegerQ[s/n], Print[n]], {n, 1, 10^8}]

PARI

list(lim)=my(v=List(),s); for(k=1,lim, s+=eulerphi(k); if(s%k==0, listput(v, k))); Vec(v) \\ Charles R Greathouse IV, Feb 07 2017

Formula

Sum_{k=1..m} phi(k) is about (3/Pi^2)*m^2 [cf. A002088, first formula].

Not obviously infinite; rough heuristics predict about 3/2 log(N) terms less than N, log(N) even ones and log(N)/2 odd ones.

Extensions

10 more terms computed by Dean Hickerson

One more term from Robert G. Wilson v, Sep 07 2001

More terms from Naohiro Nomoto, Mar 22 2002

5 more terms from Jud McCranie, Jun 21 2005