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A048290 Numbers n such that n divides Sum_{k=1..n} phi(k). 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Donovan Johnson, Table of n, a(n) for n = 1..37 (terms < 10^12)

Bender, Patashnik and Rumsey, Pizza Slicing, Phi's and the Riemann Hypothesis, American Mathematical Monthly, Vol. 101 (1994), pp. 307-317.

D. Rusin, Euler phi function

FORMULA

The sum to n is about (3/Pi^2)*n^2.

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

EXAMPLE

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

MAPLE

with(numtheory);

A048290:=proc(q) local a, n; a:=0;

for n from 1 to q do

a:=a+phi(n); if type(a/n, integer) then print(n); fi; od; end:

A048290(10^10); # Paolo P. Lava, Mar 27 2013

MATHEMATICA

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

PROG

(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

CROSSREFS

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

Sequence in context: A037079 A101325 A042980 * A306885 A029939 A082198

Adjacent sequences:  A048287 A048288 A048289 * A048291 A048292 A048293

KEYWORD

nonn,nice

AUTHOR

David J. Rusin

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

STATUS

approved

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Last modified July 11 13:25 EDT 2020. Contains 335626 sequences. (Running on oeis4.)