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Length of the longest subsequence of 1, ..., n on which the number of divisors function tau A000005 is nondecreasing.
12

%I #22 Sep 17 2023 21:22:02

%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 14,15,15,15,15,15,16,17,17,17,18,19,19,20,20,20,20,20,20,21,21,21,21,

%U 22,22,23,23,24,24,24,24,25,25,25,25,26,26,27,27,27,27

%N Length of the longest subsequence of 1, ..., n on which the number of divisors function tau A000005 is nondecreasing.

%C The sequence was inspired by A365339.

%H Peter Luschny, <a href="/A365399/b365399.txt">Table of n, a(n) for n = 1..10000</a>

%H Plot2, <a href="https://oeis.org/plot2a?name1=A365399&amp;name2=A365339&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">A365399 vs A365339</a>.

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

%e The terms of the subsequences of A000005 are marked by '*'. They start:

%e 1*, 2, 2 , 3, 2, 4, 2, 4, ... -> a(1) = 1

%e 1*, 2*, 2 , 3, 2, 4, 2, 4, ... -> a(2) = 2

%e 1*, 2*, 2*, 3, 2, 4, 2, 4, ... -> a(3) = 3

%e 1*, 2*, 2*, 3*, 2, 4, 2, 4, ... -> a(4) = 4

%e 1*, 2*, 2*, 3*, 2, 4, 2, 4, ... -> a(5) = 4

%e 1*, 2*, 2*, 3*, 2, 4*, 2, 4, ... -> a(6) = 5

%e 1*, 2*, 2*, 3*, 2, 4*, 2, 4, ... -> a(7) = 5

%e 1*, 2*, 2*, 3*, 2, 4*, 2, 4*, ... -> a(8) = 6

%e Example: a(2000000) = 450033.

%o (Julia)

%o # Computes the first N terms of the sequence using function tau from A000005.

%o function LLS_list(seq, N)

%o lst = zeros(Int64, N)

%o dyn = zeros(Int64, N)

%o for n in 1:N

%o p = seq(n)

%o nxt = dyn[p] + 1

%o while p <= N && dyn[p] < nxt

%o dyn[p] = nxt

%o p += 1

%o end

%o lst[n] = dyn[n]

%o end

%o return lst

%o end

%o A365399List(N) = LLS_list(tau, N)

%o println(A365399List(69))

%o (Python)

%o from bisect import bisect

%o from sympy import divisor_count

%o def A365399(n):

%o plist, qlist, c = tuple(divisor_count(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 04 2023

%Y Cf. A000005, A365339, A365398.

%K nonn

%O 1,2

%A _Peter Luschny_, Sep 03 2023