A076310 - OEIS

%I #32 Jan 17 2025 11:00:48

%S 0,4,8,12,16,20,24,28,32,36,1,5,9,13,17,21,25,29,33,37,2,6,10,14,18,

%T 22,26,30,34,38,3,7,11,15,19,23,27,31,35,39,4,8,12,16,20,24,28,32,36,

%U 40,5,9,13,17,21,25,29,33,37,41,6,10,14,18,22,26,30,34,38,42,7,11,15,19,23

%N a(n) = floor(n/10) + 4*(n mod 10).

%C (n==0 modulo 13) iff (a(n)==0 modulo 13); applied recursively, this property provides a divisibility test for numbers given in base 10 notation.

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

%H Reinhard Zumkeller, <a href="/A076310/b076310.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>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,1,-1).

%F a(n) = +a(n-1) +a(n-10) -a(n-11). G.f.: -x*(-4-4*x-4*x^2-4*x^3-4*x^4-4*x^5-4*x^6-4*x^7-4*x^8+35*x^9) / ( (1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) *(x-1)^2 ). - _R. J. Mathar_, Feb 20 2011

%e 435598 is not a multiple of 13, as 435598 -> 43559+4*8=43591 -> 4359+4*1=4363 -> 436+4*3=448 -> 44+4*8=76 -> 7+4*6=29=13*2+3, therefore the answer is NO.

%e Is 8424 divisible by 13? 8424 -> 842+4*4=858 -> 85+4*8=117 -> 11+4*7=39=13*3, therefore the answer is YES.

%p A076310:=n->floor(n/10) + 4*(n mod 10); seq(A076310(n), n=0..100); # _Wesley Ivan Hurt_, Jan 30 2014

%t Table[Floor[n/10] + 4*Mod[n, 10], {n, 0, 100}] (* _Wesley Ivan Hurt_, Jan 30 2014 *)

%t LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,4,8,12,16,20,24,28,32,36,1},80] (* _Harvey P. Dale_, Sep 30 2015 *)

%o (Haskell)

%o a076310 n =  n' + 4 * m where (n', m) = divMod n 10

%o -- _Reinhard Zumkeller_, Jun 01 2013

%o (PARI) a(n) = n\10 + 4*(n % 10); \\ _Michel Marcus_, Jan 31 2014

%o (Magma) [Floor(n/10)+4*(n mod 10): n in [0..75]]; // _Vincenzo Librandi_, Feb 27 2016

%Y Cf. A008595, A076309, A076311, A076312.

%K nonn,changed

%O 0,2

%A _Reinhard Zumkeller_, Oct 06 2002