The DP structure itself wasn't the hard part, that clicked pretty fast.
Recognize this as a dynamic programming problem where dp[i] = number of ways to reach step i, computed as dp[i] = dp[i-1] + sum(dp[i-p]) for all valid prime p ending in 3. Precompute all such primes up to n using a sieve, then iterate i from 1 to n, using modulo 1,000,000,007 to keep numbers manageable.
Pro tip: Mention that you can optimize memory to O(max_prime) by keeping a sliding window of recent dp values, and emphasize that the sieve is O(n log log n) while the DP is O(n * number_of_primes), which is efficient for n=100,000.
Clarify that we need to count distinct sequences of jumps, where each jump is either 1 or a prime ending in 3. Define dp[i] as the number of ways to reach step i exactly.
Generate all primes up to n using the Sieve of Eratosthenes, then filter those whose decimal representation ends in 3 (i.e., p % 10 == 3).
For each i from 1 to n, dp[i] = dp[i-1] + sum(dp[i-p]) for all valid primes p ≤ i. Use modulo 1,000,000,007 at each addition.
Iterate i from 1 to n, compute dp[i] using the recurrence. Optionally, use a sliding window or prefix sums to reduce memory, but O(n) space is acceptable for n=100,000.
Time complexity: O(n log log n) for sieve + O(n * k) for DP, where k is the number of valid primes (≈ n / (10 log n)). Space: O(n). Handle n=0 (return 1) and ensure modulo operations prevent overflow.
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