[seqfan] Re: N for which Landau's function doesn't give a signature leader

[Sean A. Irvine]

I couldn't run Chai Wah Wu's Python program directly because my Python
install is not new enough to have lcm, but I can run the following in
Python 3.8:

from math import gcd
from functools import lru_cache

@lru_cache(maxsize=None)
def A000793(n): return 1 if n == 0 else max(int((i*A000793(n-i))/gcd(i,
A000793(n-i))) for i in range(1, n+1))

A000793(29)

2310 [correct]

However:

A000793(40)

20020 [incorrect]

Sean.


On Sat, 8 Oct 2022 at 21:30, Don Reble via SeqFan <seqfan at list.seqfan.eu>
wrote:

>     I doubt Chai Wah Wu's Python program for A000793.
>     Can someone test whether it computes a(29) correctly?
>
> > The exceptional values [landau(n) isn't a signature leader] of n begin
> > 3, 8, 9, 14, 15, 16, 25, 26, 27, 32, 33, 38, 39, 42, 57 ...
>     landau(25), landau(26) are signature leaders.
>
> > Research questions:
>     Some questions can be answered by scanning the b-files.
>     (But I haven't verified those.)
>
> > Are there an infinite number of exceptions?
>     The "exceptional" values seem to be more common than the rest.
>     Of the first 40001 values, only 8772 are signature leaders (A025487).
>
> > do all the exceptions consist of a signature leader times one or two
> > larger primes?
>     landau(640) is a 53-smooth signature leader times 61*67*73.
>     The extra primes needn't be consecutive.
>
>     The extra primes can be small.
>     landau(36548) is a 647-smooth signature leader times 19*659.
>
> --
> Don Reble  djr at nk.ca
>
>
>
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>