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A371156
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Length of the longest subsequence of 1, ..., n on which the Dedekind psi function (A001615) is nondecreasing.
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1
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1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 25, 26, 26, 27, 27, 27, 27, 28, 28, 29, 29, 29, 29, 30, 30, 31, 31, 31, 32, 33, 33, 34, 34, 34
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OFFSET
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1,2
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COMMENTS
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The envelope max_{i<=n} (a(i)-A000720(i)) appears to be slowly increasing as n increases. For instance, a(1)-A000720(1)=1, whereas a(374598)-A000720(374598)=91 and a(642852)-A000720(642852)=96.
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LINKS
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FORMULA
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0 <= a(n+1) - a(n) <= 1.
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EXAMPLE
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a(7) = 6 because A001615 is nondecreasing on 1,2,3,4,5,6 or 1,2,3,4,5,7 but not on 1,2,3,4,5,6,7.
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PROG
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(Python)
from math import prod
from bisect import bisect
from sympy import primefactors
def f(n):
r = primefactors(n)
return n*prod(p+1 for p in r)//prod(r)
plist, qlist, c = tuple(f(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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