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

%I #10 May 24 2020 17:38:13

%S 1,2,6,30,210,2310,30030,120120,690690,1021020,2042040,4084080,

%T 9699690,58198140,96996900,106696590,223092870,892371480,6469693230,

%U 6915878970,19409079690,32348466150,71166625530,200560490130,7420738134810,8624101075590

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

%C Somayajulu (1950) proved that phi(k+1)/phi(k) is unbounded, hence this sequence is infinite.

%C a(27) <= 16445960190660. - _Giovanni Resta_, May 24 2020

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

%D B. S. K. R. Somayajulu, On Euler's totient function phi(n), Math. Student, Vol. 18 (1950), pp. 31-32.

%e The values of phi(k+1)/phi(k) for the first terms are 1, 2, 3, 3.75, 4.375, 4.8125, ...

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

%Y Cf. A000010, A335070.

%K nonn,more

%O 1,2

%A _Amiram Eldar_, May 22 2020

%E a(24)-a(26) from _Giovanni Resta_, May 24 2020