A335070 - OEIS

A335070

Numbers k where records occur for phi(k)/phi(k+1), where phi is the Euler totient function (A000010).

%I #14 Jan 09 2025 10:34:31

%S 1,5,11,17,23,29,59,89,149,179,209,389,419,839,1049,1259,1889,2099,

%T 2309,9239,11549,13859,20789,23099,25409,30029,90089,180179,210209,

%U 270269,300299,330329,390389,420419,510509,570569,1021019,2042039,3063059,4084079,4594589

%N Numbers k where records occur for phi(k)/phi(k+1), where phi is the Euler totient function (A000010).

%C Somayajulu (1950) proved that phi(k+1)/phi(k) can be arbitrarily close to 0, therefore phi(k)/phi(k+1) is unbounded and this sequence is infinite.

%D József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 16.

%H Amiram Eldar, <a href="/A335070/b335070.txt">Table of n, a(n) for n = 1..75</a>

%H B. S. K. R. Somayajulu, On Euler's totient function phi(n), Math. Student, Vol. 18 (1950), pp. 31-32; <a href="https://schoolbooksarchive.azimpremjiuniversity.edu.in/handle/20.500.12497/11744">entire issues 1 and 2</a>.

%e The values of phi(k)/phi(k+1) for the first terms are 1, 2, 2.5, 2.666..., 2.75, 3.5, ...

%t rm = 0; s1 = 1; seq = {}; Do[s2 = EulerPhi[n]; If[(r = s1/s2) > rm, rm = r; AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^6}]; seq

%Y Cf. A000010, A335069.

%K nonn

%O 1,2

%A _Amiram Eldar_, May 22 2020