A365738 -id:A365738 - cp's OEIS frontend

A365742 Length of the largest subset of 1,...,n on which the Euler totient function phi A000010 is constant.

1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10

Offset: 1

Chai Wah Wu, Sep 17 2023

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000
R. C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arith. 83 (1998), no. 4, 331-361.
Paul Erdős, On the normal number of prime factors of p - 1 and some related problems concerning Euler’s phi-function, Quart J. Math 6 (1935), 205-213.
Paul Pollack, Carl Pomerance and Enrique Treviño, Sets of monotonicity for Euler's totient function, preprint. See M0(n).
Paul Pollack, Carl Pomerance and Enrique Treviño, Sets of monotonicity for Euler's totient function, Ramanujan J. 30 (2013), no. 3, 379--398.
Terence Tao, Monotone non-decreasing sequences of the Euler totient function, arXiv:2309.02325 [math.NT], 2023.

Cf. A000010, A000720.

Cf. A365398, A365399, A365400, A365474, A365737, A365738, A365740, A365741, A061070.

from collections import Counter
from sympy import totient
def A365742(n): return max(Counter(totient(i) for i in range(1,n+1)).values())

Pollack et al. showed that A365737(n)-a(n) > n^0.18 for large n.

A365748 a(n) = A365742(10^n).

Original entry on oeis.org

1, 3, 10, 30, 72, 247, 937, 2844, 9261, 30742

Offset: 0

Chai Wah Wu, Sep 17 2023

R. Baker and G. Harman, Shifted primes without large prime factors, Acta Arithmetica 83 (1998), pp. 331-361.
Paul Pollack, Carl Pomerance, and Enrique Treviño, Sets of monotonicity for Euler's totient function, preprint. See M0(n).
Paul Pollack, Carl Pomerance, and Enrique Treviño, Sets of monotonicity for Euler's totient function, Ramanujan J. 30 (2013), no. 3, 379-398.
Terence Tao, Monotone non-decreasing sequences of the Euler totient function, arXiv:2309.02325 [math.NT], 2023.

Cf. A000010, A000720.

Cf. A365398, A365399, A365400, A365474, A365737, A365738, A365740, A365741, A365742, A061070.

from collections import Counter
from sympy import totient
def A365748(n): return max(Counter(totient(i) for i in range(1,10**n+1)).values())

Baker and Harman showed that a(n) >= 10^(0.7038n) for all large enough n. - Chai Wah Wu, Oct 17 2023

cp's OEIS Frontend

A365742 Length of the largest subset of 1,...,n on which the Euler totient function phi A000010 is constant.

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