A257079 The least nonzero digit missing from the factorial representation (A007623) of n.

1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 3, 1, 3, 3, 3, 1, 3, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 4, 4, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 3, 1, 4, 4, 4, 1, 4, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 4, 4, 1, 4, 4, 4, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 3, 1, 3, 3, 3, 1, 3, 1, 2, 2, 2, 1, 5, 2

Offset

0,2

Links

Antti Karttunen, Table of n, a(n) for n = 0..10080

Eric Angelini, et al., "Multiply by the fantom digit", Discussion on Seqfan-list.

Formula

Other identities:

For all n >= 1, a(A033312(n)) = n. [n! - 1 gives the first position where n appears. Note also how the digits in factorial base representation may get arbitrarily large values.]

Example

The least digit > 0 missing from the factorial representation (A007623) of zero, "0", is 1, thus a(0) = 1.
The least digit > 0 missing from the factorial representation of one, "1", is 2, thus a(1) = 2.
The least digit > 0 missing from the factorial representation of 21, "311", is 2, thus a(21) = 2.

Mathematica

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Min[Complement[Range[Max[s]+1], s]]]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, Jan 24 2024 *)

Prog

(Scheme)
(define (A257079 n) (let loop ((digs (uniq (sort (n->factbase n) <))) (mnp 1)) (cond ((null? digs) mnp) ((zero? (car digs)) (loop (cdr digs) mnp)) ((= (car digs) mnp) (loop (cdr digs) (+ 1 mnp))) (else mnp))))
;; Convert an integer to a factorial expansion list:
(define (n->factbase n) (let loop ((n n) (fex (if (zero? n) (list 0) (list))) (i 2)) (cond ((zero? n) fex) (else (loop (floor->exact (/ n i)) (cons (modulo n i) fex) (1+ i))))))
(define (uniq lista) (let loop ((lista lista) (z (list))) (cond ((null? lista) (reverse! z)) ((and (pair? z) (equal? (car z) (car lista))) (loop (cdr lista) z)) (else (loop (cdr lista) (cons (car lista) z))))))

Crossrefs

Cf. A007623, A257080.

Cf. A033312 (the positions of records from a(1) onward.)

Cf. A255411 (the positions of ones.)

Sequence in context: A295277 A194290 A329028 * A327567 A260372 A037180

Adjacent sequences: A257076 A257077 A257078 * A257080 A257081 A257082

Keyword

nonn,base

Author

Antti Karttunen, Apr 15 2015

Status

approved