I tried it myself with the program at the end of this post, and got a
bit different of a result. The program printed:

times tested: 100000000, sum: 1600092924, average: 16.0009
times tested: 200000000, sum: 2952473665, average: 14.7624
times tested: 300000000, sum: 4316535702, average: 14.3885
times tested: 400000000, sum: 6018669380, average: 15.0467
times tested: 500000000, sum: 7585216727, average: 15.1704
times tested: 600000000, sum: 8915924298, average: 14.8599
times tested: 700000000, sum: 10662688388, average: 15.2324
times tested: 800000000, sum: 12027713479, average: 15.0346
times tested: 900000000, sum: 13468996092, average: 14.9656
times tested: 1000000000, sum: 15300481793, average: 15.3005

  The program:

#include "IsaacRand.hh"
#include <iostream>
#include <cassert>

int main()
{
    typedef unsigned long long ULL;

    IsaacRand rng(0);

    ULL sum = 0;
    std::cout << "sizeof(sum) = " << sizeof(sum) << std::endl;

    ULL counter;
    for(counter = 0; counter < 1000000000ULL; ++counter)
    {
        ULL rounds = 0;
        while(rng.getNext() % 2 == 0)
        {
            ++rounds;
            assert(rounds < 64);
        }
        if(rounds) sum += 1ULL << rounds;
        else ++sum;

        if(counter % 100000000 == 100000000-1)
            std::cout << "times tested: " << counter+1
                      << ", sum: " << sum << ", average: "
                      << double(sum)/double(counter) << std::endl;
    }
}

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