[seqfan] A new digital problem

[Vladimir Shevelev]

 Dear SeqFans, 
 
I am interested in the following problem.
 
 Let g_1,g_2,...,g_r be integers more than 1. Denote G=2*Prod_{i=1,...,r}(g_i-1). Let m be positive integer. Denote s_i(m) sum of digits of mG in base g_i. Then numbers s_i/(g_i-1) are integers.
Does exist, for given integers  2<=g_1<g_2<...<g_r, any number m=m(g_1,...,g_r) such that all numbers s_i/(g_i-1), i=1,...,r, are even?
   For example, for g_1=2, g_2=4, g_3=5, we have G=24 and the minimal requied m is m=5. Indeed, we have 120_2=1111000, 120_4=1320, 120_5=440. So, s_1(5)/1=4, s_2(5)/3=2, s_3(5)/4=2.
  In particular, sequences of the following type are of interest:
 
  a(n) be the smallest m 1) for r=1, g_1=n, n>=2; 2)  for r=2, g_1=2, g_2=n;, n>=3; 3) for r=3, g_1=2, g_2=3, g_3=n, n>=4; 4) for r=4, g_1=2, g_2=3, g_3=4, g_4=n, n>=5, etc.
 
Regards,
Vladimir

 Shevelev Vladimir‎