A380341 a(n) = phi(1 + phi(4 + phi(9 + ... phi(n^2)))).

1, 2, 4, 6, 6, 6, 16, 16, 22, 16, 16, 16, 16, 22, 22, 16, 22, 16, 16, 16, 16, 16, 16, 22, 16, 22, 16, 16, 16, 22, 22, 22, 22, 22, 22, 16, 16, 22, 16, 22, 16, 16, 16, 16, 16, 16, 22, 16, 16, 16, 16, 16, 22, 16, 16, 16, 16, 16, 22, 16, 16, 16, 16, 16, 16, 16, 16, 16, 22

Offset

1,2

Links

Paolo Xausa, Table of n, a(n) for n = 1..1000

Luis Palacios Vela and Christian Wolird, The Forestry of Adversarial Totient Iterations, arXiv:2501.10616 [math.NT], 2025.

Index entries for linear recurrences with constant coefficients, signature (1).

Formula

From Paolo Xausa, Jan 22 2025: (Start)

a(n) = 1 for n = 1;

a(n) = 2 for n = 2;

a(n) = 4 for n = 3;

a(n) = 6 for n = 4, 5, 6;

a(n) = 22 for n = 9, 14, 15, 17, 24, 26, 30, 31, 32, 33, 34, 35, 38, 40, 47, 53, 59, 69;

a(n) = 16 otherwise (see Vela and Wolird). (End)

G.f.: x*(6*x^69 - 6*x^68 + 6*x^59 - 6*x^58 + 6*x^53 - 6*x^52 + 6*x^47 - 6*x^46 + 6*x^40 - 6*x^39 + 6*x^38 - 6*x^37 + 6*x^35 - 6*x^29 + 6*x^26 - 6*x^25 + 6*x^24 - 6*x^23 + 6*x^17 - 6*x^16 + 6*x^15 - 6*x^13 + 6*x^9 - 6*x^8 - 10*x^6 - 2*x^3 - 2*x^2 - x - 1)/(x - 1). - Chai Wah Wu, Jan 22 2025

Mathematica

A380341[n_] := Fold[EulerPhi[#2 + #] &, 0, Range[n, 1, -1]^2]; Array[A380341, 100] (* or *)
A380341[n_] := Which[n <= 2, n, n == 3, 4, 4 <= n <= 6, 6, MemberQ[{9, 14, 15, 17, 24, 26, 30, 31, 32, 33, 34, 35, 38, 40, 47, 53, 59, 69}, n], 22, True, 16]; Array[A380341, 100] (* Paolo Xausa, Jan 22 2025 *)

Prog

(PARI) a(n) = my(x=0); forstep(k=n, 1, -1, x = eulerphi(x+k^2)); x;
(Python)
from functools import reduce
from sympy import totient
def A380341(n): return reduce(lambda x, y:totient(x)+y, (m**2 for m in range(n, -1, -1))) # Chai Wah Wu, Jan 22 2025

Crossrefs

Cf. A000010, A380340, A380342, A380354, A380414, A380415.

Sequence in context: A078611 A211376 A278249 * A339743 A131450 A114218

Adjacent sequences: A380338 A380339 A380340 * A380342 A380343 A380344

Keyword

nonn,easy

Author

Michel Marcus, Jan 22 2025

Status

approved