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Numbers n such that the sum of the Euler totient functions of integers up to n is a square.
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%I #23 Sep 21 2017 04:22:37

%S 1,3,14,32,54,1458,3765,5343,10342,57918,72432,134072,1103584,4984175,

%T 9191040,18399460,49034273,176485286,423360893,1432766906,62342171276,

%U 433015422290,1192983964547,2034643004727,2742734055027

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

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

%C a(22) > 10^11. - _Donovan Johnson_, Mar 15 2011

%C a(26) > 3*10^12. - _Giovanni Resta_, Sep 21 2017

%H Javier Cilleruelo and Florian Luca, <a href="http://digital.csic.es/bitstream/10261/31070/1/Sum%2520of%2520primes.pdf">On the sum of the first n primes</a>, Q. J. Math. 59:4 (2008), 14 pp.

%H Florian Luca and Ayyadurai Sankaranarayanan, <a href="http://www.smm.org.mx/boletin_anterior/v14/n1.pdf">On numbers n such that phi(1)+...+phi(n) is a square</a>, Bol. Soc. Mat. Mexicana (3), Vol. 14 (2008), pp. 1-6.

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

%e 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.

%t T = 0; For[c = 1, c < 1000000, c++, T = T + EulerPhi[l]; If[T == Floor[Sqrt[T]]^2, Print[c, " ", Floor[Sqrt[T]]]]] (* Luca *)

%t searchMax = 2000; phiRunSum = Accumulate[EulerPhi[Range[searchMax]]]; Select[Range[searchMax], IntegerQ[Sqrt[phiRunSum[[#]]]] &] (* _Alonso del Arte_, Sep 19 2017 *)

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

%Y Cf. A000010, A002088.

%K nonn

%O 1,2

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

%E a(13)-a(19) from _Donovan Johnson_, Dec 02 2009

%E a(20)-a(21) from _Donovan Johnson_, Mar 15 2011

%E a(22)-a(25) from _Giovanni Resta_, Sep 21 2017