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Shift factorial base representation left by one digit.
49

%I #31 Apr 27 2020 02:32:50

%S 0,2,6,8,12,14,24,26,30,32,36,38,48,50,54,56,60,62,72,74,78,80,84,86,

%T 120,122,126,128,132,134,144,146,150,152,156,158,168,170,174,176,180,

%U 182,192,194,198,200,204,206,240,242,246,248,252,254,264,266,270,272

%N Shift factorial base representation left by one digit.

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

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

%H Antti Karttunen, <a href="/A153880/b153880.txt">Table of n, a(n) for n = 0..40319</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%F Other identities. For all n >= 0:

%F A266193(a(n)) = n.

%e 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.

%t 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 *)

%o (Scheme)

%o (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)))))))

%o (Python)

%o from sympy import factorial as f

%o def a007623(n, p=2): return n if n<p else a007623((n//p), p+1)*10 + n%p

%o def a(n):

%o x = (str(a007623(n)) + '0')[::-1]

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

%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 20 2017

%Y Indices of zeros in A260736.

%Y Cf. A153883 (terms divided by 2).

%Y Cf. A266193 (a left inverse).

%Y Cf. A273670 (complement).

%Y Cf. also A007623, A225901, A255411.

%K base,nonn

%O 0,2

%A _Antti Karttunen_, Jan 03 2009