A365398 Length of the longest subsequence of 1, ..., n on which sigma, the sum of the divisors of n (A000203), is nondecreasing.

1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 24, 24, 25, 25, 25

Offset: 1

Author

Peter Luschny, Sep 08 2023

Keywords

• nonn

Comments

The sequence was inspired by A365339. In particular, note remark (4.4) by Terence Tao in the linked paper.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Plot2, A365398 vs A365339.
• Terence Tao, Monotone non-decreasing sequences of the Euler totient function, arXiv:2309.02325 [math.NT], 2023.

Crossrefs

Cf. A000203, A000720, A365339, A365399, A365397.

Cf. A061069.

Programs

Python

from bisect import bisect
from sympy import divisor_sigma
def A365398(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 c # Chai Wah Wu, Sep 08 2023

Formula

a(n+1) - a(n) <= 1.

a(n) >= A000720(n)+1 since A000203(p) = p+1 for p prime. - Chai Wah Wu, Sep 08 2023