kill(all);

van=facebook.com/LearningCalculus$

      Problem 1:

f(i):=is(integerp(sqrt(i)));

a0:2;

create_list(if f(i) then sqrt(i) else i+3, i, a0, 25);

create_list(if f(i) then     (1) else (0), i, a0, 25);

a0: 4$

create_list(if f(i) then 1 else 0, i, a0, 26);

      There is no a0 such that
      there is a  number A such that an = A for infinitely many values of n,
      since the distribution of numbers that have an integer square root
      is infinite.
      They can be generated by squaring the integers in succession infinitely.
      These number perforate the square-root series at an ever increasing spacing.
      That spacing increases linearly as seen above where the zeroes appear in
      stretches of 2, 4, 6, 8 zeros respectively.
      The i+3 members are specious entries that do not contribute to the result.