A153880 - OEIS

A153880

Shift factorial base representation left by one digit.

0, 2, 6, 8, 12, 14, 24, 26, 30, 32, 36, 38, 48, 50, 54, 56, 60, 62, 72, 74, 78, 80, 84, 86, 120, 122, 126, 128, 132, 134, 144, 146, 150, 152, 156, 158, 168, 170, 174, 176, 180, 182, 192, 194, 198, 200, 204, 206, 240, 242, 246, 248, 252, 254, 264, 266, 270, 272

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OFFSET

0,2

COMMENTS

Equally, append 0 to the end of the factorial base representation of n (= A007623(n)), then convert back to decimal.

Involution A225901 maps each term of this sequence to a unique term of A255411, and vice versa.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..40319

Index entries for sequences related to factorial base representation

FORMULA

Other identities. For all n >= 0:

A266193(a(n)) = n.

EXAMPLE

Factorial base representation of 5 is A007623(5) = "21". Shifting this once left (that is, appending 0 to the end) yields "210", which is factorial base representation for 14. Thus a(5) = 14.

MATHEMATICA

Table[Function[b, FromDigits[IntegerDigits[n, b]~Join~{0}, b]]@ MixedRadix[Reverse@ Range@ 12], {n, 0, 57}] (* Michael De Vlieger, May 30 2016, Version 10.2 *)

PROG

(Scheme)

(define (A153880 n) (let loop ((n n) (z 0) (i 2) (f 2)) (cond ((zero? n) z) (else (loop (floor->exact (/ n i)) (+ (* f (modulo n i)) z) (+ 1 i) (* 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)) + '0')[::-1]

return 0 if n==0 else sum(int(x[i])*f(i + 1) for i in range(len(x)))

print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

CROSSREFS

Indices of zeros in A260736.

Cf. A153883 (terms divided by 2).

Cf. A266193 (a left inverse).

Cf. A273670 (complement).