A116943 - OEIS

%I #6 Oct 31 2013 12:17:38

%S 0,0,1,0,1,0,1,0,0,1,3,2,2,2,1,1,3,4,5,3,3,1,4,4,7,2,7,7,4,6,9,9,6,5,

%T 5,7,4,9,4,7,7,7,10,8,6,8,6,9,8,9,8,10,11,11,8,13,5,11,15,13,10,10,8,

%U 12,9,14,11,8,11,12,10,13,13,13,10,10,12,6,10,15,8,17,17,16,16,12,16,15,13

%N Number of 4s digits plus non-final 3s digits 3 base 5 expansion of 2^n.

%C In his comment on A038003 Frank Adams-Watters conjectures "that 2^n contains such a base 5 digit for n>=9. This is almost certainly true." That is equivalent to a(n) > 0 for n>=9, which is also equivalent to A094389(n) = 5 where A094389 is last decimal digit of the odd Catalan number A038003(n).

%e a(7) = 0 because 2^7 (modulo 5) = 1003, which contains 0 digits 4 plus 0 non-final digits 3 (it has a digit 3, but that digit is finial, meaning rightmost).

%e a(10) = 3 because 2^10 mod 5 = 13044, which contains 2 digits 4 plus 1 non-final digits 3, so 2 + 1 = 3.

%e a(60) = 10 because 2^60 mod 5 = 34132411211412413323100401, which contains 5 digits 4 plus 5 non-final digits 3, so 5 + 5 = 10.

%t f[n_] := Block[{id = IntegerDigits[2^n, 5]}, Count[id, 4] + Count[Most@id, 3]]; Table[ f[n], {n, 0, 88}] (* _Robert G. Wilson v_ *)

%Y Cf. A038003, A094389.

%K base,easy,nonn

%O 0,11

%A _Jonathan Vos Post_, Mar 23 2006

%E More terms from _Robert G. Wilson v_, Apr 01 2006