A376702 Numbers k whose nonzero digits are strictly decreasing when written in factoradic.

1, 2, 4, 5, 6, 12, 13, 14, 18, 19, 20, 22, 23, 24, 48, 49, 50, 54, 72, 73, 74, 76, 77, 78, 84, 85, 86, 96, 97, 98, 100, 101, 102, 108, 109, 110, 114, 115, 116, 118, 119, 120, 240, 241, 242, 246, 264, 360, 361, 362, 364, 365, 366, 372, 373, 374, 384, 408, 409, 410

Offset: 1

Author

Douglas Boffey, Oct 02 2024

Keywords

• nonn
• base

Comments

Base factorial is defined as the right hand digit being the units, the next left being the 2's, then the 6's, and so on.

Examples

			50, being 2010 (base !), is included, whereas 51, being 2011 (base !), is not included.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Wikipedia, Factorial number system

Crossrefs

Cf. A007623, A108731.

Programs

PARI

isok(n)={my(k=1,p=0); while(n, k++; my(r=n%k); if(r, if(r<=p, return(0)); p=r); n\=k); 1} \\ Andrew Howroyd, Oct 04 2024

Python

def f(n, i=2): return [n] if n < i else [*f(n//i, i=i+1), n%i]
def ok(n):
    fnz = [d for d in f(n) if d != 0]
    return len(fnz) == len(set(fnz)) and fnz == sorted(fnz, reverse=True)
print([k for k in range(1, 411) if ok(k)]) # Michael S. Branicky, Oct 02 2024

Python

# faster for initial segment of sequence
from math import factorial
from itertools import count, islice
def bgen(d, i): # strictly decreasing non-zero elmts <= i and dth digit from left <= d
    if d < 1: yield tuple(); return
    yield from ((j,) + t for j in range(0, min(i+1, d+1)) for t in bgen(d-1, i if j == 0 else j-1))
def agen(): # generator of terms
    for digits in count(1):
        for first in range(1, digits+1):
            for rest in bgen(digits-1, first-1):
                t = (first, ) + rest
                yield sum(factorial(i)*d for i, d in enumerate(t[::-1], 1))
print(list(islice(agen(), 60))) # Michael S. Branicky, Oct 02 2024