A351987 Numbers with factorial base expansion digits in nonincreasing order.

0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 30, 32, 33, 48, 54, 56, 57, 60, 62, 63, 64, 65, 72, 78, 80, 81, 84, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 102, 104, 105, 108, 110, 111, 112, 113, 114, 116, 117, 118, 119, 120, 144, 150

Offset: 1

Author

Rémy Sigrist, Feb 27 2022

Keywords

• nonn
• base

Comments

This sequence is to factorial base what A009996 is to decimal base.

Examples

			The factorial base expansion of 102 is "4100", so 102 belongs to this sequence.
The factorial base expansion of 103 is "4101", so 103 does not belong to this sequence.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..6910 (terms <= 9!)
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A009996, A108731, A351988.

Programs

Mathematica

max = 6; q[n_] := AllTrue[Differences @ IntegerDigits[n, MixedRadix[Range[max, 2, -1]]], # <= 0 &]; Select[Range[0, max!], q] (* Amiram Eldar, Feb 28 2022 *)

PARI

is(n) = { my (p=0); for (r=2, oo, if (n==0, return (1)); my (d=n%r); if (d
				

Python

def facbase(n, i=2): return [n] if n < i else [*facbase(n//i, i=i+1), n%i]
def ok(n): return (fb:=facbase(n)) == sorted(fb, reverse=True)
print([k for k in range(151) if ok(k)]) # Michael S. Branicky, Mar 09 2025

Python

# faster for initial segment of sequence
from math import factorial
from itertools import count, islice, product
def bgen(d, first, last): # generator of non-increasing factorial base tuples
    yield from ((i,) + t for i in range(first, last+1) for t in bgen(d-1, first=0, last=min(i, d-1))) if d else (tuple(),)
def A351987_gen(): # generator of terms
    yield from (sum(dj*factorial(j) for j, dj in enumerate(t[::-1], 1)) for d in count(0) for t in bgen(d, 1, d))
print(list(islice(A351987_gen(), 63))) # Michael S. Branicky, Mar 09 2025