A268864 - OEIS

%I #22 Apr 02 2016 23:33:27

%S 1,1,1,3,2,2,8,4,6,52,9,8,34,11,15,47,18,22,195,23,9,77,32,22,57,9,15,

%T 160,11,11,0,7,4,0,3,0,4,6,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Number of integers of the form n*k whose digits sum up to n where 0 <= k <= n^2.

%C It appears that a(n)=0 for n>39. This has been checked for n<100.

%C If n has d digits, i.e. 10^(d-1) <= n < 10^d, then k*n <= n^3 has at most 3*d digits, so its sum of digits is at most 27*d. But 27*d < 10^(d-1) if d >= 3. Thus all terms for n >= 100 are 0. - _Robert Israel_, Mar 20 2016

%e When n = 4, the digit sum of 4*k only equals 4 when k = 1 and k = 10; thus a(4) = 2.

%p f:=  n -> nops(select(t -> convert(convert(n*t,base,10),`+`)=n,[$0..n^2])):

%p map(f, [$0..100]); # _Robert Israel_, Mar 20 2016

%o (PARI) a(n) = sum(k=0, n^2, sumdigits(k*n)==n); \\ _Michel Marcus_, Feb 15 2016

%K nonn,base

%O 0,4

%A _Jordan G. Fischer_, Feb 15 2016