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%I #21 Sep 10 2023 13:14:46
%S 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,
%T 14,15,15,15,15,15,15,16,16,16,16,17,17,18,18,18,18,18,18,19,19,19,19,
%U 19,19,20,20,21,21,21,21,22,22,22,23,24,24,25,25,25
%N Length of the longest subsequence of 1, ..., n on which sigma, the sum of the divisors of n (A000203), is nondecreasing.
%C The sequence was inspired by A365339. In particular, note remark (4.4) by Terence Tao in the linked paper.
%H Chai Wah Wu, <a href="/A365398/b365398.txt">Table of n, a(n) for n = 1..10000</a>
%H Plot2, <a href="https://oeis.org/plot2a?name1=A365398&name2=A365339&tform1=untransformed&tform2=untransformed&shift=0&radiop1=ratio&drawlines=true">A365398 vs A365339</a>.
%H Terence Tao, <a href="https://arxiv.org/abs/2309.02325">Monotone non-decreasing sequences of the Euler totient function</a>, arXiv:2309.02325 [math.NT], 2023.
%F a(n+1) - a(n) <= 1.
%F a(n) >= A000720(n)+1 since A000203(p) = p+1 for p prime. - _Chai Wah Wu_, Sep 08 2023
%o (Python)
%o from bisect import bisect
%o from sympy import divisor_sigma
%o def A365398(n):
%o plist, qlist, c = tuple(divisor_sigma(i) for i in range(1,n+1)), [0]*(n+1), 0
%o for i in range(n):
%o qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i
%o c = max(c,a)
%o return c # _Chai Wah Wu_, Sep 08 2023
%Y Cf. A000203, A000720, A365339, A365399, A365397.
%Y Cf. A061069.
%K nonn
%O 1,2
%A _Peter Luschny_, Sep 08 2023
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