A381770 a(n) is the least k > 0 such that the factorial base expansion of k*n has digits in nonincreasing order.
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 6, 3, 2, 2, 3, 1, 2, 1, 1, 3, 3, 2, 3, 3, 2, 2, 14, 2, 2, 2, 2, 2, 2, 1, 6, 3, 2, 2, 8, 1, 2, 1, 1, 2, 2, 1, 5, 1, 1, 1, 1, 4, 8, 4, 6, 6, 8, 1, 6, 4, 2, 2, 9, 1, 8, 1, 1, 7, 8, 1, 5, 1
Offset: 0
Examples
The first terms, alongside the factorial base expansion of n*a(n), are: n a(n) fact(n*a(n)) -- ---- ------------ 0 1 0 1 1 1 2 1 1,0 3 1 1,1 4 1 2,0 5 1 2,1 6 1 1,0,0 7 2 2,1,0 8 1 1,1,0 9 1 1,1,1 10 2 3,1,0 11 2 3,2,0 12 1 2,0,0 13 5 2,2,2,1 14 1 2,1,0 15 1 2,1,1
Links
Programs
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PARI
is(n) = { my (p = -1); for (r = 2, oo, if (n==0, return (1), p > p = n%r, return (0)); n \= r;); } a(n) = { for (k = 1, oo, if (is(k*n), return (k););); } -
Python
from itertools import count def facbase(n, i=2): return [n] if n < i else [*facbase(n//i, i=i+1), n%i] def is_non_inc(n): return (fb:=facbase(n)) == sorted(fb, reverse=True) def a(n): return next(k for k in count(1) if is_non_inc(k*n)) print([a(n) for n in range(87)]) # Michael S. Branicky, Mar 09 2025
Formula
a(n) <= (n-1)! for any n > 0.
a(n) = 1 iff n belongs to A351987.
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