A061069 -id:A061069 - cp's OEIS frontend

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

Peter Luschny, Sep 08 2023

nonn

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

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.

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

Cf. A061069.

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

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

A385991 a(n) is the number of distinct values among A002487(0), ..., A002487(n).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16

Offset: 0

Rémy Sigrist, Jul 14 2025

nonn

This sequence exhibits large runs of consecutive equal values.

			Sequence begins:
  n   a(n)  A002487(n)
  --  ----  ----------
   0     1           0
   1     2           1
   2     2           1
   3     3           2
   4     3           1
   5     4           3
   6     4           2
   7     4           3
   8     4           1
   9     5           4
  10     5           3
  11     6           5
  12     6           2
  13     6           5
  14     6           3
  15     6           4
  16     6           1
  17     6           5
  18     6           4
  19     7           7

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program
Index entries for sequences related to Stern's sequences

See A061069, A061070 and A061071 for similar sequences.

Cf. A002487, A091945, A385993.

PARI
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def A385991(n):
    if n==0: return 1
    a, b, s, c = 0, 1, {0,1}, 2
    for i in range(n-1):
        a, b = b, ((a//b<<1)+1)*b-a
        if b not in s:
            c += 1
            s.add(b)
    return c # Chai Wah Wu, Jul 17 2025

a(A091945(n)) = n (this is the first occurrence of n in the sequence).

a(2*n) = a(2*n-1) for any n > 0.

cp's OEIS Frontend

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

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