A342702 Indices of records of A007015.

1, 2, 4, 6, 12, 18, 24, 30, 48, 60, 78, 90, 120, 150, 180, 210, 330, 360, 390, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 3360, 3570, 3990, 4200, 4620, 5460, 6300, 6930, 9240, 10710, 10920, 11550, 13860, 16380, 17220, 17850, 18480, 20790, 27720, 30030, 39270

Offset: 1

Author

Amiram Eldar, Mar 18 2021

Keywords

• nonn

Comments

Numbers m such that the smallest solution k to the equation phi(m+k) = phi(k) is larger than all the corresponding smallest solutions for all numbers below m.

The corresponding record values are 1, 4, 8, 24, 48, 52, 96, ... (see the link for more values).

Apparently, a(n) is even for n > 1, divisible by 6 for n > 3, by 30 for n > 9, and by 210 for n > 19. These observations are based on data up to n=100.

It seems that in general, for all k >= 1 there is a number n_k such that all the terms a(n) with n > n_k are divisible by the first k primes.

Furthermore, it seems that all the terms are of the form m*p^e, were m is a term of A055932, and p^e is a prime power (A000961).

Examples

			The first 6 terms of A007015 are 1, 4, 3, 8, 5 and 24. The record values, 1, 4, 8 and 24 occur at 1, 2, 4 and 6, the first 4 terms of this sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..100
• Amiram Eldar, Table of n, a(n), A007015(a(n)) for n = 1..100

Crossrefs

Cf. A000961, A007015, A055932.

Programs

Mathematica

f[n_] := Module[{k = 1}, While[EulerPhi[n + k] != EulerPhi[k], k++]; k]; fm =0; s = {}; Do[f1 = f[n];  If[f1 > fm, fm = f1; AppendTo[s, n]];, {n, 1, 1000}]; s