A257684 Discard the rightmost digit from the factorial base representation of n and subtract one from all remaining nonzero digits, then convert back to decimal.

0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17

Offset

0,13

Comments

In other words, subtract one from all nonzero digits in the factorial base representation (A007623) of n and shift it one step right (i.e., delete the rightmost zero), then convert back to decimal.

Links

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

Formula

For all n >= 0, a(A255411(n)) = n. [This sequence works as a left inverse of A255411. See A257685 for a "cleaned up" version.]

Example

For 4, whose factorial base representation is "20" (as 4 = 2*2! + 0*1!), when we discard the rightmost zero, and subtract 1 from 2, we get "1", thus a(4) = 1.
For 18, whose factorial base representation is "300" (as 18 = 3*3! + 0*2! + 0*1!), when we discard the rightmost zero, and subtract 1 from 3, we get "20", thus a(18) = 4.

Mathematica

nn = 95; m = 1; While[Factorial@ m < nn, m++]; m; Map[FromDigits[#, MixedRadix[Reverse@ Range[2, m]]] &[If[# == 0, 0, # - 1] & /@ Most@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]]] &, Range[0, nn]] (* Michael De Vlieger, Aug 11 2016, Version 10.2 *)

Prog

(Scheme) (define (A257684 n) (let loop ((n n) (z 0) (i 2) (f 0)) (cond ((zero? n) z) (else (let ((d (remainder n i))) (loop (quotient n i) (+ z (* f (- d (if (zero? d) 0 1)))) (+ 1 i) (if (zero? f) 1 (* f (- i 1)))))))))
(Python)
from sympy import factorial as f
def a007623(n, p=2):
    return n if n<p else a007623(n//p, p+1)*10 + n%p
def a(n):
    x=str(a007623(n))[:-1]
    y="".join(str(int(i) - 1) if int(i)>0 else '0' for i in x)[::-1]
    return 0 if n==1 else sum(int(y[i])*f(i + 1) for i in range(len(y)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

Crossrefs

Positions of zeros: A059590.

Cf. A007623, A255411, A257685, A257687.

Can be used to define simple recurrences for sequences like A034968, A246359, A257679, A257694, A257695 and A257696.

Sequence in context: A024542 A355880 A209082 * A098424 A098428 A023193

Adjacent sequences: A257681 A257682 A257683 * A257685 A257686 A257687

Keyword

nonn,base

Author

Antti Karttunen, May 04 2015

Status

approved