%I #14 Jan 02 2016 04:13:13

%S 1,0,0,1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,

%T 0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0,

%U 0,1,0,2,0,0,0,0,1,1,0,1,1,0,0,2,0,0,0,1,0,2,1,0,0,0,0,2,0,1,1,1,0,1,0,1,1,0,0,2,0,1,0,2,0,1,0,0,1,0,1,3

%N a(n) = number of ways n can be divided into two factors that have the same number of digits in factorial base representation (the two different orders for unequal factors are counted only once).

%H Antti Karttunen, <a href="/A266344/b266344.txt">Table of n, a(n) for n = 1..14161</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%F a(n) = Sum_{d|n} [(d <= (n/d)) and (A084558(d) = A084558(n/d))].

%F (In the above formula [ ] stands for Iverson bracket, giving as its result 1 only if d is less than or equal to n/d and in factorial base representation d and n/d require equal number of digits, and 0 otherwise.)

%e 1 can be factored just one way, as 1*1, and thus a(1) = 1.

%e 4 can be factored as 2*2, and thus also a(4) = 1, and generally for all perfect squares k, a(k) >= 1.

%e 14 can be factored as 2*7, but as A007623(2) = 2 and A007623(7) = 101, with different number of digits in factorial base (and 1*14 fares even less well), a(14) = 0.

%e 72 can be factored to two divisors so that the factorial base representations are of equal length as 6*12 or 8*9 (where the corresponding factorial base representations are "100" * "200" and "110" * "111"), thus a(72) = 2.

%e 120 can be similarly factored as 6*20 ("100" * "310"), 8*15 ("110" * "211") and 10*12 ("120" * "200"), thus a(120) = 3.

%t r = Most@ Reverse@ Range@ 10; Map[Length, Table[Flatten@ Map[Differences@ IntegerLength[#, MixedRadix@ r] &, Transpose@ {#, n/#}] &@ TakeWhile[Divisors@ n, # <= Sqrt@ n &], {n, 120}] /. k_ /; k > 0 -> Nothing] (* _Michael De Vlieger_, Dec 30 2015, Version 10.2 *)

%o (PARI)

%o A084558(n) = { my(m=1); if(0==n,n,while(m!<=n,m++);return(m-1)); }

%o A266344(n) = sumdiv(n, d, ((d <= (n/d)) && (A084558(d)==A084558(n/d))));

%o for(n=1, 14161, write("b266344.txt", n, " ", A266344(n)));

%Y Cf. A084558.

%Y Cf. A266345 (positions of records).

%Y Cf. also A078781, A266342.

%K nonn,base

%O 1,72

%A _Antti Karttunen_, Dec 28 2015