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%I #12 Jun 13 2017 19:18:02
%S 2,2,6,4,80,6,46802,8,468,80,68024,24,468,46802,4680,80,68,468,6802,
%T 80,80246802468,68024,46,24,0,468,680246802,80246802468,680246802468,
%U 4680,24680246802468,0,680246802468024,68,802468024680,468,24680246802468,6802
%N Smallest multiple of n with property that digits are even and each digit is two more (mod 10) than the previous digit; or 0 if no such multiple exists.
%C If a(n) = 0, a(n*k) = 0 for any positive k. - _Franklin T. Adams-Watters_, Nov 03 2009
%C 25 is impossible; its multiples end either with the digits 00 or 50.
%C Multiples of 16 except 16 and 80 are impossible. Of the 625 multiples of 16 mod 10000, none are 246, 2468, 4680, 6802, or 8024. That leaves only 80 as a possible value for multiples of 16. It appears that the multiples of 16 and 25 are the only numbers for which a(n) = 0 - _Franklin T. Adams-Watters_, Nov 03 2009
%H Franklin T. Adams-Watters, <a href="/A062400/b062400.txt">Table of n, a(n) for n = 1..1000.</a>
%e a(7) = 7*6686 = 46802 and this number has increasing larger even digits (mod 10). a(12) = 24 = 12*2 has increasing even digits.
%t f[n_] := Block[{x = 0, a = IntegerDigits[n], i = 1}, l =Length[a]; While < l, If[ Mod[ a[[i]] + 2, 10] != a + 1]], x = 1]; i++ ]; Return[x]]; Dock = n; While[ Union[ even[ IntegerDigits[k]]] != {True} || Fmk] == 1, k += n]; Print[k], {n, 1, 20}]
%o (PARI) evenincr(n)=local(d,r);d=n%4*2+2;n\=4;r=0;for(k=0,n,r=r*10+(d+2*k)%10);r
%o a(n)=if(n%25==0 || (n%16==0&80%n!=0), 0, k=0; while(evenincr(k)%n!=0, k++); evenincr(k)) /* This program will loop if the conjecture above is incorrect. */
%K nonn,base,easy
%O 1,1
%A _Amarnath Murthy_, Jun 28 2001
%E Edited and extended by _Robert G. Wilson v_, Feb 22 2002
%E Edited and extended by _Franklin T. Adams-Watters_, Nov 03 2009