{"app":{"locale":"en-GB"},"problem":{"content":"

Consider the Gaussian integer i-1. A base i-1 representation of a Gaussian integer a+bi is a finite sequence of digits d_n-1d_n-2...d₁d₀ such that:

a+bi = d_n-1(i-1)^n-1 + d_n-2(i-1)^n-2 + ... + d₁(i-1) + d₀
Each d_k is in {0,1}
There are no leading zeroes, i.e. d_n-1 ≠ 0, unless a+bi is itself 0

Here are base i-1 representations of a few Gaussian integers:

11+24i → 111010110001101
24-11i → 110010110011
8+0i → 111000000
-5+0i → 11001101
0+0i → 0

Remarkably, every Gaussian integer has a unique base i-1 representation!

Define f(a+bi) as the number of 1s in the unique base i-1 representation of a+bi. For example, f(11+24i) = 9 and f(24-11i) = 7.

Define B(L) as the sum of f(a+bi) for all integers a, b such that |a| ≤ L and |b| ≤ L. For example, B(500) = 10795060.

Find B(10¹⁵) mod 1 000 000 007.","id":508,"title":"Integers in base i-1"},"problems":[{"id":1,"title":"Multiples of 3 and 5"},{"id":2,"title":"Even Fibonacci numbers"},{"id":3,"title":"Largest prime factor"},{"id":4,"title":"Largest palindrome product"},{"id":5,"title":"Smallest multiple"},{"id":6,"title":"Sum square difference"},{"id":7,"title":"10001st prime"},{"id":8,"title":"Largest product in a series"},{"id":9,"title":"Special Pythagorean triplet"},{"id":10,"title":"Summation of primes"},{"id":11,"title":"Largest product in a grid"},{"id":12,"title":"Highly divisible triangular number"},{"id":13,"title":"Large sum"},{"id":14,"title":"Longest Collatz sequence"},{"id":15,"title":"Lattice paths"},{"id":16,"title":"Power digit sum"},{"id":17,"title":"Number letter counts"},{"id":18,"title":"Maximum path sum I"},{"id":19,"title":"Counting Sundays"},{"id":20,"title":"Factorial digit sum"},{"id":21,"title":"Amicable numbers"},{"id":22,"title":"Names scores"},{"id":23,"title":"Non-abundant sums"},{"id":24,"title":"Lexicographic permutations"},{"id":25,"title":"1000-digit Fibonacci number"},{"id":26,"title":"Reciprocal cycles"},{"id":27,"title":"Quadratic primes"},{"id":28,"title":"Number spiral diagonals"},{"id":29,"title":"Distinct powers"},{"id":30,"title":"Digit fifth powers"},{"id":31,"title":"Coin sums"},{"id":32,"title":"Pandigital products"},{"id":33,"title":"Digit cancelling fractions"},{"id":34,"title":"Digit factorials"},{"id":35,"title":"Circular primes"},{"id":36,"title":"Double-base palindromes"},{"id":37,"title":"Truncatable primes"},{"id":38,"title":"Pandigital multiples"},{"id":39,"title":"Integer right triangles"},{"id":40,"title":"Champernowne's constant"},{"id":41,"title":"Pandigital prime"},{"id":42,"title":"Coded triangle numbers"},{"id":43,"title":"Sub-string divisibility"},{"id":44,"title":"Pentagon numbers"},{"id":45,"title":"Triangular, pentagonal, and hexagonal"},{"id":46,"title":"Goldbach's other conjecture"},{"id":47,"title":"Distinct primes factors"},{"id":48,"title":"Self powers"},{"id":49,"title":"Prime permutations"},{"id":50,"title":"Consecutive prime sum"},{"id":51,"title":"Prime digit replacements"},{"id":52,"title":"Permuted multiples"},{"id":53,"title":"Combinatoric selections"},{"id":54,"title":"Poker hands"},{"id":55,"title":"Lychrel numbers"},{"id":56,"title":"Powerful digit sum"},{"id":57,"title":"Square root convergents"},{"id":58,"title":"Spiral primes"},{"id":59,"title":"XOR decryption"},{"id":60,"title":"Prime pair sets"},{"id":61,"title":"Cyclical figurate numbers"},{"id":62,"title":"Cubic permutations"},{"id":63,"title":"Powerful digit counts"},{"id":64,"title":"Odd period square roots"},{"id":65,"title":"Convergents of e"},{"id":66,"title":"Diophantine equation"},{"id":67,"title":"Maximum path sum II"},{"id":68,"title":"Magic 5-gon ring"},{"id":69,"title":"Totient maximum"},{"id":70,"title":"Totient permutation"},{"id":71,"title":"Ordered 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