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

Allan Wechsler acwacw at gmail.com
Fri Oct 7 02:25:05 CEST 2022

Landau's function <https://oeis.org/A000793> gives the highest possible
least common multiple for any partition of n.

Signature leaders <https://oeis.org/A025487> are the least possible number
with a given prime-power signature; their factorization consists of
consecutive prime numbers starting with 2, and with exponents never

For reasons that I hope are intuitively obvious, but which I can't easily
verbalize, one expects Landau's function to yield signature leaders a lot
of the time. However, it does deliver non-leaders quite a lot as well.

The exceptional values of n begin: 3, 8, 9, 14, 15, 16, 25, 26, 27, 32, 33,
38, 39, 42, 57 ...

Research questions: Are there an infinite number of exceptions? After a few
"very exceptional" small quirks, do all the exceptions consist of a
signature leader times one or two larger primes? Do exceptions get rarer?

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