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

1,3

Comments

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

Links

Rémy Sigrist, Table of n, a(n) for n = 1..6910 (terms <= 9!)

Index entries for sequences related to factorial base representation

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A009996, A108731, A351988.

Sequence in context: A165707 A052063 A129525 * A295165 A258023 A190232

Adjacent sequences: A351984 A351985 A351986 * A351988 A351989 A351990

Keyword

nonn,base

Author

Rémy Sigrist, Feb 27 2022

Status

approved