A076309 - OEIS

%I #26 May 25 2019 01:52:42

%S 0,-2,-4,-6,-8,-10,-12,-14,-16,-18,1,-1,-3,-5,-7,-9,-11,-13,-15,-17,2,

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

%U -2,-4,-6,-8,-10,-12,-14,5,3,1,-1,-3,-5,-7,-9,-11,-13,6,4,2,0,-2,-4,-6,-8,-10

%N a(n) = floor(n/10) - 2*(n mod 10).

%C Delete the last digit from n and subtract twice this digit from the shortened number. - _N. J. A. Sloane_, May 25 2019

%C (n==0 modulo 7) iff (a(n)==0 modulo 7); applied recursively, this property provides a useful test for divisibility by 7.

%D Erdős, Paul, and János Surányi. Topics in the Theory of Numbers. New York: Springer, 2003. Problem 6, page 3.

%D Karl Menninger, Rechenkniffe, Vandenhoeck & Ruprecht in Goettingen (1961), 79A.

%H Reinhard Zumkeller, <a href="/A076309/b076309.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisibilityTests.html">Divisibility Tests</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisibility_rule">Divisibility rule</a>

%F From _R. J. Mathar_, Nov 23 2010: (Start)

%F a(n) = a(n-1) + a(n-10) - a(n-11).

%F G.f.: x*(-2 -2*x -2*x^2 -2*x^3 -2*x^4 -2*x^5 -2*x^6 -2*x^7 -2*x^8 +19*x^9)/((1+x)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)*(x-1)^2). (End)

%e 695591 is not a multiple of 7, as 695591 -> 69559-2*1=69557 -> 6955-2*7=6941 -> 694-2*1=692 -> 69-2*2=65=7*9+2, therefore the answer is NO.

%e Is 3206 divisible by 7? 3206 -> 320-2*6=308 -> 30-2*8=14=7*2, therefore the answer is YES, indeed 3206=2*7*229.

%t Table[Floor[n/10] - 2*Mod[n, 10], {n, 0, 100}] (* _G. C. Greubel_, Apr 07 2016 *)

%o (Haskell)

%o a076309 n =  n' - 2 * m where (n', m) = divMod n 10

%o -- _Reinhard Zumkeller_, Jun 01 2013

%o (PARI) a(n) = n\10 - 2*(n % 10); \\ _Michel Marcus_, Apr 07 2016

%Y Cf. A008589, A076310, A076311, A076312.

%K sign

%O 0,2

%A _Reinhard Zumkeller_, Oct 06 2002