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
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
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