A365397 a(n) = 64 + A000720(n) - A365398(n).

63, 63, 63, 62, 63, 62, 63, 62, 62, 61, 62, 61, 62, 62, 61, 60, 61, 60, 61, 60, 60, 60, 61, 60, 60, 60, 60, 59, 60, 59, 60, 60, 60, 60, 60, 59, 60, 60, 60, 59, 60, 59, 60, 60, 60, 60, 61, 60, 60, 60, 60, 60, 61, 60, 60, 59, 59, 59, 60, 59, 60, 60, 59, 58, 58

Offset

1,1

Comments

It is conjectured that A365339(n) - PrimePi(n) = 64 for all n >= 31957 (Pollack et al.). Does a similar relation apply if one replaces Euler's totient by the sum of divisors function in A365339? In particular, note remark (4.4) by Terence Tao in the linked paper.

From Chai Wah Wu, Sep 08 2023: (Start)

a(n) seems to be decreasing for n=10^i:

a(1) = 63

a(10) = 61

a(100) = 58

a(1000) = 58

a(10^4) = 54

a(10^5) = 53

a(10^6) = 48

a(10^7) = 46

a(10^8) = 43

(End)

Links

Table of n, a(n) for n=1..65.

Paul Pollack, Carl Pomerance, and Enrique Treviño, Sets of monotonicity for Euler's totient function, preprint. See M(n).

Paul Pollack, Carl Pomerance, and Enrique Treviño, Sets of monotonicity for Euler's totient function, Ramanujan J. 30 (2013), no. 3, pp. 379-398.

Terence Tao, Monotone non-decreasing sequences of the Euler totient function, arXiv:2309.02325 [math.NT], 2023.

Formula

a(n)<=63. This is due to the fact that A000203(p) = p+1 for p prime, and therefore A365398(n) >= A000720(n)+1. - Chai Wah Wu, Sep 08 2023

Prog

(Python)
from bisect import bisect
from sympy import divisor_sigma, primepi
def A365397(n):
    plist, qlist, c = tuple(divisor_sigma(i) for i in range(1, n+1)), [0]*(n+1), 0
    for i in range(n):
        qlist[a:=bisect(qlist, plist[i], lo=1, hi=c+1, key=lambda x:plist[x])]=i
        c = max(c, a)
    return 64+primepi(n)-c # Chai Wah Wu, Sep 08 2023

Crossrefs

Cf. A000720, A000203, A365398, A365400.

Sequence in context: A278827 A125638 A365400 * A332926 A112816 A296875

Adjacent sequences: A365394 A365395 A365396 * A365398 A365399 A365400

Keyword

nonn

Author

Peter Luschny, Sep 08 2023

Status

approved