A130697 Numbers n such that the sum of the Euler totient functions of integers up to n is a square.

1, 3, 14, 32, 54, 1458, 3765, 5343, 10342, 57918, 72432, 134072, 1103584, 4984175, 9191040, 18399460, 49034273, 176485286, 423360893, 1432766906, 62342171276, 433015422290, 1192983964547, 2034643004727, 2742734055027

Offset

1,2

Comments

Luca & Sankaranarayanan show that this sequence is of asymptotic density zero. It is not known if the sequence is infinite.

a(22) > 10^11. - Donovan Johnson, Mar 15 2011

a(26) > 3*10^12. - Giovanni Resta, Sep 21 2017

Links

Table of n, a(n) for n=1..25.

Javier Cilleruelo and Florian Luca, On the sum of the first n primes, Q. J. Math. 59:4 (2008), 14 pp.

Florian Luca and Ayyadurai Sankaranarayanan, On numbers n such that phi(1)+...+phi(n) is a square, Bol. Soc. Mat. Mexicana (3), Vol. 14 (2008), pp. 1-6.

Formula

Numbers n such that phi(1) + phi(2) + ... + phi(n) = x^2 with some integer x.

Example

a(3) = 14 since phi(1) + phi(2) + phi(3) + phi(4) + phi(5) + phi(6) + phi(7) + phi(8) + phi(9) + phi(10) + phi(11) + phi(12) + phi(14) = 8^2.

Mathematica

T = 0; For[c = 1, c < 1000000, c++, T = T + EulerPhi[l]; If[T == Floor[Sqrt[T]]^2, Print[c, " ", Floor[Sqrt[T]]]]] (* Luca *)
searchMax = 2000; phiRunSum = Accumulate[EulerPhi[Range[searchMax]]]; Select[Range[searchMax], IntegerQ[Sqrt[phiRunSum[[#]]]] &] (* Alonso del Arte, Sep 19 2017 *)

Prog

(PARI) s=0; for(n=1, 1e7, if(issquare(s+=eulerphi(n)), print1(n", "))) \\ Charles R Greathouse IV, Feb 01 2013

Crossrefs

Cf. A000010, A002088.

Sequence in context: A032525 A294420 A197946 * A033991 A155154 A081269

Adjacent sequences: A130694 A130695 A130696 * A130698 A130699 A130700

Keyword

nonn

Author

Florian Luca (fluca(AT)matmor.unam.mx), Jul 11 2007

Extensions

a(13)-a(19) from Donovan Johnson, Dec 02 2009

a(20)-a(21) from Donovan Johnson, Mar 15 2011

a(22)-a(25) from Giovanni Resta, Sep 21 2017

Status

approved