

A109012


a(n) = gcd(n,9).


5



9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1
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OFFSET

0,1


COMMENTS

Start with positive integer n. At each step, either (a) multiply by any positive integer or (b) remove all zeros from the number. a(n) is the smallest number that can be reached by this process.  David W. Wilson, Nov 01 2005
From Martin Fuller, Jul 09 2007: (Start)
Also the minimal positive difference between numbers whose digit sum is a multiple of n. Proof:
Construction: Pick a positive number that does not end with 9, and has a digit sum na(n). To form the lower number, append 9 until the digit sum is a multiple of n. This is always possible since the difference is gcd(n,9). Add a(n) to form the higher number, which will have digit sum n.
E.g., n=12: prefix=18, lower=18999, higher=19002, difference=3.
Minimality: All numbers are a multiple of a(n) if their digit sum is a multiple of n. Hence the minimal difference is at least a(n). (End)


LINKS

Table of n, a(n) for n=0..100.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1)


FORMULA

a(n) = 1 + 2*[3n] + 6*[9n], where [xy] = 1 when x divides y, 0 otherwise.
a(n) = a(n9).
Multiplicative with a(p^e, 9) = gcd(p^e, 9).  David W. Wilson Jun 12 2005
G.f.: (9  x  x^2  3*x^3  x^4  x^5  3*x^6  x^7  x^8) / ((x1)*(1 + x + x^2)*(x^6 + x^3 + 1)).  R. J. Mathar, Apr 04 2011
Dirichlet g.f.: (1+2/3^s+6/9^s)*zeta(s).  R. J. Mathar, Apr 04 2011


PROG

(PARI) a(n)=gcd(n, 9) \\ Charles R Greathouse IV, Oct 07 2015
(Python)
from math import gcd
def a(n): return gcd(n, 9)
print([a(n) for n in range(101)]) # Michael S. Branicky, Sep 01 2021


CROSSREFS

Cf. A109004.
Cf. A109007, A109008, A109009, A109010, A109011, A109013, A109014, A109015.
Sequence in context: A010163 A200128 A225537 * A037478 A198551 A010162
Adjacent sequences: A109009 A109010 A109011 * A109013 A109014 A109015


KEYWORD

nonn,easy,mult


AUTHOR

Mitch Harris


STATUS

approved



