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Sum of Digit Sequence

Let a_0, a_1, a_2,... be an integer sequence defined by: a_0 = 1. a_n = a_(n-1) + s(a_(n-1)) for n >= 1, where s(m) denotes the sum of the base-10 digits of m. The first terms are 1, 2, 4, 8, 16, 2...

Source sync Apr 19, 2026
Problem #0551
Level Level 22
Solved By 546
Languages C++, Python
Answer 73597483551591773
Length 474 words
dynamic_programmingsequencemodular_arithmetic

Problem Statement

This archive keeps the full statement, math, and original media on the page.

Let \(a_0, a_1, \dots \) be an integer sequence defined by:

  • \(a_0 = 1\);

  • for \(n \le 1\), \(a_n\) is the sum of the digits of all preceding terms.

The sequence starts with \(1, 1, 2, 4, 8, 16, 23, 28, 38, 49, \dots \)

You are given \(a_{10^6} = 31054319\).

Find \(a_{10^{15}}\).

Problem 551: Sum of Digit Sequence

Mathematical Foundation

Definition 1. For a non-negative integer mm with base-10 representation m=j=0d1mj10jm = \sum_{j=0}^{d-1} m_j \cdot 10^j where 0mj90 \leq m_j \leq 9, the digit sum is s(m)=j=0d1mjs(m) = \sum_{j=0}^{d-1} m_j.

Theorem 1 (Digit-Sum Growth Bound). For all n0n \geq 0, if ana_n has dd digits, then 1s(an)9d1 \leq s(a_n) \leq 9d. Consequently, an=O(nlogn)a_n = O(n \log n).

Proof. Since an1a_n \geq 1 for all n0n \geq 0 (by induction, as a0=1a_0 = 1 and each step adds a positive digit sum), every digit representation is non-trivial and s(an)1s(a_n) \geq 1. Each digit is at most 9, so s(an)9ds(a_n) \leq 9d where d=log10(an)+1d = \lfloor \log_{10}(a_n) \rfloor + 1. The recurrence gives an=a0+k=0n1s(ak)a_n = a_0 + \sum_{k=0}^{n-1} s(a_k), and since s(ak)9(log10(ak)+1)=O(logak)s(a_k) \leq 9(\lfloor \log_{10}(a_k) \rfloor + 1) = O(\log a_k), standard analysis yields an=O(nlogn)a_n = O(n \log n). In particular, a1015a_{10^{15}} has at most 17 digits. \square

Lemma 1 (Block Invariance). Write ana_n in positional blocks of width BB: decompose an=c0+c110B+c2102B+a_n = c_0 + c_1 \cdot 10^B + c_2 \cdot 10^{2B} + \cdots where 0cj<10B0 \leq c_j < 10^B. Define supper=j1s(cj)s_{\mathrm{upper}} = \sum_{j \geq 1} s(c_j), the digit sum contributed by all blocks above the lowest. Then:

(i) The full digit sum satisfies s(an)=s(c0)+suppers(a_n) = s(c_0) + s_{\mathrm{upper}}.

(ii) If c0+s(c0)+supper<10Bc_0 + s(c_0) + s_{\mathrm{upper}} < 10^B (no carry out of the lowest block), then the transition anan+1a_n \mapsto a_{n+1} leaves all upper blocks unchanged.

(iii) As a consequence, suppers_{\mathrm{upper}} remains constant across all steps in which no carry propagates out of the lowest block.

Proof. Part (i) follows from the additivity of digit sums across independent blocks. For (ii), adding s(an)=s(c0)+suppers(a_n) = s(c_0) + s_{\mathrm{upper}} to ana_n modifies only the lowest block: c0c0+s(c0)+supperc_0 \mapsto c_0 + s(c_0) + s_{\mathrm{upper}}, and the no-carry hypothesis ensures this remains below 10B10^B, leaving c1,c2,c_1, c_2, \ldots intact. Part (iii) is immediate from (ii). \square

Theorem 2 (Block Transition Function). For block size BB, define the transition

T(V,U)=(Δ,V)T(V, U) = (\Delta, V')

where V{0,,10B1}V \in \{0, \ldots, 10^B - 1\} is the current lowest-block value, U0U \geq 0 is the upper digit sum, Δ\Delta is the number of sequence steps until the first carry propagates out of the block, and VV' is the new lowest-block value after those Δ\Delta steps. Then:

(i) TT is well-defined: starting from any (V,U)(V, U) with V+U1V + U \geq 1, a carry must eventually occur.

(ii) Δ10B\Delta \leq 10^B, and in practice Δ=O(10B/(U+1))\Delta = O(10^B / (U + 1)).

(iii) V=(Vfinal+s(Vfinal)+U)10BV' = (V_{\mathrm{final}} + s(V_{\mathrm{final}}) + U) - 10^B, where VfinalV_{\mathrm{final}} is the block value at the last step before carry.

Proof. For (i), by Lemma 1(iii), UU is constant until carry. Each step increments VV by s(V)+U1s(V) + U \geq 1 (since an1a_n \geq 1 ensures s(V)+U1s(V) + U \geq 1). Hence VV strictly increases and must eventually reach or exceed 10B10^B. For (ii), the minimum per-step increment is 1, giving Δ10B\Delta \leq 10^B; when UU is large the average increment is roughly U+O(logV)U + O(\log V), so Δ10B/U\Delta \approx 10^B / U. Part (iii) follows directly from the definition. \square

Corollary 1 (Memoizability). The function T(V,U)T(V, U) depends only on (V,U)(V, U). With B=4B = 4, the domain is {0,,9999}×{0,,916}\{0, \ldots, 9999\} \times \{0, \ldots, 9 \cdot 16\}, giving a table of 104×1451.45×10610^4 \times 145 \approx 1.45 \times 10^6 entries that can be lazily computed and cached.

Editorial

The algorithm maintains ana_n as an array of 20 base-10 digits and uses block-based acceleration with memoization. We use dynamic programming over the state space implied by the derivation, apply each admissible transition, and read the answer from the final table entry.

Pseudocode

repeat
else

Complexity Analysis

  • Table construction: At most 104×1451.45×10610^4 \times 145 \approx 1.45 \times 10^6 entries, each requiring O(104)O(10^4) steps in the worst case but O(104/(U+1))O(10^4 / (U+1)) on average.
  • Main loop: Each block jump advances the sequence by Θ(104/U)\Theta(10^4 / U) steps on average, so the number of outer iterations is O(NUavg/104)O(N \cdot U_{\mathrm{avg}} / 10^4). With N=1015N = 10^{15} and typical average U70U \approx 70, the outer loop executes on the order of 101310^{13} iterations in the worst case but much fewer due to caching and carry propagation patterns. Total runtime is several minutes.
  • Space: O(104×Umax)O(1.5×106)O(10^4 \times U_{\max}) \approx O(1.5 \times 10^6) for the memo table.

Answer

73597483551591773\boxed{73597483551591773}

Code

Each problem page includes the exact C++ and Python source files from the local archive.

C++ project_euler/problem_551/solution.cpp
#include <bits/stdc++.h>
using namespace std;

/*
 * Problem 551: Sum of Digits Sequence
 * a_0 = 1, a_n = a_{n-1} + digitsum(a_{n-1})
 * Find a_{10^15}.
 *
 * Approach: Block-based acceleration.
 * We split the number into blocks of digits and precompute jump tables.
 * For a given low-order block value, we know how many steps until a carry
 * propagates out of that block, and the total digit-sum contribution.
 *
 * We use a multi-level approach with blocks of B digits each.
 */

typedef long long ll;
typedef unsigned long long ull;

// We'll represent the current number as an array of digits (or blocks)
// and use a recursive approach to jump through steps.

// digit sum of a number
int digitsum(ll n) {
    int s = 0;
    while (n > 0) {
        s += n % 10;
        n /= 10;
    }
    return s;
}

int main() {
    ios_base::sync_with_stdio(false);

    const ll TARGET = 1000000000000000LL; // 10^15

    // Simple approach using block acceleration:
    // We process digit-by-digit with jump tables.
    // For each "level" (ones, tens, hundreds, etc.), we precompute
    // how many steps and how much total digit-sum increment happens
    // before a carry propagates to the next level.

    // Level 0: single digit d (0-9)
    // steps[0][d] = number of steps until d rolls over past 9
    // delta[0][d] = total increment (sum of digit sums) over those steps
    // But this is tricky because the digit sum depends on ALL digits.

    // Better approach: The digit sum contribution from digit position k is
    // just that digit's value. So digitsum(a) = sum of all digits.
    // When we add digitsum(a) to a, the low-order digits change,
    // and carries may propagate.

    // We use the following approach:
    // Maintain the number as a vector of digits.
    // For each "block" (individual digit), precompute:
    //   Given digit value d and a "carry-in" rate c (the digit sum of higher digits),
    //   how many steps until this digit overflows (causes carry to next position)?

    // Actually, let's use a well-known approach for this problem:
    // We define for each suffix length L:
    //   jump(L, suffix_value) = (steps, new_suffix_value, carry_count)
    // where steps = how many a-sequence steps until carry propagates past position L,
    // and carry_count = how many carries propagate (0 or 1 per step group).

    // For this problem, we use a simpler direct computation with jump tables.
    // Reference approach: process in blocks where we track the bottom k digits.

    // Actually, the most practical approach for this problem:
    // The number has at most ~17 digits for a_{10^15}.
    // We process one digit position at a time from least significant.
    //
    // For position p (0-indexed from right), we track:
    //   - how many steps we can take before digit at position p overflows
    //   - the accumulated effect on lower digits
    //
    // Since the digit sum = sum of all digits, and adding it affects the number:
    // The key insight is that when we add the digit sum, the contribution from
    // digits at positions >= p+1 is constant (they don't change until carry reaches them).

    // Let's use a direct simulation with memoization per digit level.
    // We'll represent the state as (bottom_digits_value, higher_digitsum)
    // At each level k, we merge steps until the k-th digit changes.

    // Practical implementation:
    // We keep the number in an array of 20 digits.
    // For each level from 0 to 19, we precompute (using lower-level tables):
    //   Given (lower_value, upper_dsum), compute (steps, new_lower_value, carries)
    //   where carries = number of carries propagated past this level.

    // Level-0: digit d, upper_dsum = S (sum of all other digits)
    //   Each step adds (d + S) to the number. This changes d to (d + (d+S)) mod 10
    //   with carry. But S also includes lower... wait, d IS the lowest digit at level 0.
    //   Actually for level 0, the full digit sum includes d and all higher digits (sum = S+d).
    //   Adding S+d to the number: the ones digit d becomes (d + (S+d)) mod 10 = (2d+S) mod 10.
    //   Carry out = (d + (S+d)) / 10 = (2d+S) / 10 which is 0 or 1.
    //   But wait, the NEXT step, d has changed, and also potentially higher digits changed
    //   (from carry). This gets complicated.

    // Simpler: direct simulation with acceleration.
    // At each step, compute digitsum, add to number.
    // But 10^15 steps is too many.
    //
    // We use the "carry-skip" technique:
    // The digit sum is at most 9*17 = 153.
    // So a_{n+1} - a_n <= 153.
    // For the lowest digit to "cycle" through 0-9, it takes at least 1 step.
    //
    // Multi-level jumping:
    // Level L processes the bottom L+1 digits.
    // jump_L(bottom_value, upper_dsum) returns (steps, new_bottom_value, total_carry)
    //   where steps = how many sequence steps until a carry propagates past digit L.
    //   new_bottom_value = value of bottom L+1 digits after those steps
    //   total_carry = total carry out of digit L (always 1 in the "until overflow" version)

    // For level 0 (ones digit only):
    //   d = current ones digit, S = digit sum of all other digits
    //   Each step: full digit sum = d + S + (adjustment from carries below, but level 0 has no below)
    //   new_d = d + (d + S), with carries.
    //   We step until the ones digit wraps around (causes carry).
    //   But each step changes d, and S is constant (no carry has reached higher digits yet).
    //   Steps until carry: we accumulate d = d, then d' = (2d + S) mod 10, etc.
    //   until the sum exceeds 10.

    // Let's implement this properly.

    // Represent number as array of 20 digits (index 0 = ones)
    int digits[20] = {};
    digits[0] = 1; // a_0 = 1
    int num_digits = 1;

    // For a given bottom-k digits state and upper digit sum, compute forward
    // until carry propagates past digit k.
    // We use recursive tables.

    // Cache for level k: map from (bottom_value, upper_dsum) -> (steps, new_bottom, carry)
    // But the state space is huge for large k.

    // Alternative: process level by level without full memoization.
    // For level 0: given d (ones digit) and S (sum of digits at positions >= 1),
    //   step forward until carry occurs. Return (steps, new_d, carry=1).
    // For level 1: given (d1, d0) and S (sum of digits at positions >= 2),
    //   repeatedly call level-0 jumps, accumulating steps.
    //   After each level-0 jump, d1 gets a carry added.
    //   Continue until d1 itself overflows.
    //   Return (total_steps, new_d1_d0, carry=1).
    // For level k: similar, call level-(k-1) repeatedly until digit k overflows.

    // This is the standard approach. The number of calls at each level is at most 10
    // (digit goes from current value to overflow), and there are ~17 levels.
    // Total work: 10^17 -- still too much!

    // We need memoization. State at level k: (bottom_k_digits, upper_dsum).
    // bottom_k_digits can be up to 10^k, upper_dsum up to 9*(20-k) ~ 153.
    // For k small (0,1,2), this is feasible. For larger k, we note that
    // upper_dsum is bounded by ~153, so table size = 10^k * 153.
    // For k up to 6: 10^6 * 153 ~ 1.5 * 10^8 entries -- borderline.
    // For k up to 4: 10^4 * 153 ~ 1.5 * 10^6 entries -- fine.

    // Better: use variable-size blocks. E.g., blocks of 4 digits.
    // Level 0: 4 lowest digits (0-9999), upper_dsum (0-144) -> 10000*145 ~ 1.45M entries
    // Level 1: next 4 digits... but state is (8 bottom digits, upper_dsum)
    //   = 10^8 * 100 -- too large.

    // Actually, at level 1, we don't store all 8 bottom digits.
    // We store just the level-1 block (4 digits, positions 4-7) and upper_dsum.
    // After level-1 jump, the level-0 block may have changed, but we only need
    // to track the level-1 block state and how many carries propagated.

    // Wait, I think the standard approach is:
    // Process each digit position independently, from low to high.
    // At digit position p, we know the current digit d_p and S = sum of digits above p.
    // We call the level-(p-1) jump to process all changes below digit p.
    // Each such call might or might not produce a carry into digit p.
    // We accumulate until digit p overflows, then return.

    // With memoization per level, the state is (d_p, S) where S = upper_dsum
    // and the result includes how the lower digits end up. But the lower digit
    // state matters for correct continuation. Hmm.

    // Actually a cleaner approach:
    // For each level k (digit position), we define:
    //   advance(k, digits[], target_steps)
    // which advances the sequence by target_steps, modifying digits[].
    // At level k, we use the level-(k-1) function to advance until digit k overflows.
    // The key insight: we can MEMOIZE the level-k transition because it depends only on
    // digits[0..k] and sum(digits[k+1..]), and the output is the new digits[0..k] and carry.

    // For this to work, we need: state = (digits[0..k], upper_sum) -> (steps, new digits[0..k], carry)
    // Size of state space = 10^(k+1) * max_upper_sum.
    // For k=3 (4 digits), max_upper_sum = 9*16 = 144, state space = 10000 * 145 = 1.45M
    // For k=7 (8 digits), state space = 10^8 * 100 -- too large.

    // So we use a two-level hierarchy:
    // Level A: 4 low digits, memoized with upper_dsum. Table: ~1.45M entries.
    // Level B: each of the higher digits, processed with level-A jumps.
    //   No memoization needed at level B since there are few (~13) higher digits,
    //   each going through at most 10 transitions, each requiring a level-A lookup chain.

    // Actually, the simplest efficient approach for this problem uses a
    // different technique: carry propagation tracking.

    // Let me implement the digit-by-digit level approach with memoization.
    // State: (value of bottom L+1 digits, digit sum of upper digits)
    // For L = 0..3 (4 digits), memoized.
    // For L = 4..19, process digit by digit using L=3 as the base.

    // Represent number in base-10, 20 digits max
    // digits[0] = ones, digits[1] = tens, etc.

    // Memoization table for level 3 (bottom 4 digits, values 0..9999, upper_dsum 0..160)
    const int BLOCK = 4;
    const int BLOCK_MOD = 10000; // 10^BLOCK
    const int MAX_UDSUM = 9 * 17 + 1; // max upper digit sum + 1

    // For block of bottom BLOCK digits with value V and upper digit sum U:
    // step forward until carry propagates out of these BLOCK digits.
    // Return: (steps_taken, new_V, carry_out)
    // carry_out is always 1 (we stop when overflow happens)
    // But there might be MULTIPLE overflows... no, we stop at first carry out.

    // memo[V][U] = (steps, new_V)
    // We compute this on-the-fly and cache it.

    struct Entry {
        ll steps;
        int new_val;
    };

    // Table: BLOCK_MOD * MAX_UDSUM
    static Entry memo[10000][160];
    static bool computed[10000][160];
    memset(computed, 0, sizeof(computed));

    // Function to compute one entry
    auto compute_entry = [&](int V, int U) -> Entry {
        // Simulate: bottom BLOCK digits have value V, upper digit sum is U.
        // Full digit sum = digitsum(V) + U.
        // Add digit sum to V. If V >= BLOCK_MOD, carry out, return.
        ll steps = 0;
        int v = V;
        while (true) {
            int ds = digitsum(v) + U;
            int nv = v + ds;
            steps++;
            if (nv >= BLOCK_MOD) {
                // Carry out. New value of bottom digits = nv - BLOCK_MOD
                return {steps, nv - BLOCK_MOD};
            }
            v = nv;
        }
    };

    auto get_entry = [&](int V, int U) -> Entry {
        if (U >= 160) {
            // Shouldn't happen for our problem, but handle gracefully
            return compute_entry(V, U);
        }
        if (!computed[V][U]) {
            memo[V][U] = compute_entry(V, U);
            computed[V][U] = true;
        }
        return memo[V][U];
    };

    // Now process the higher digits.
    // We have digits[0..19], with digits[0..BLOCK-1] forming the bottom block.
    // digits[BLOCK..19] are the upper digits.
    // We advance until reaching TARGET steps.

    ll steps_done = 0;

    while (steps_done < TARGET) {
        // Compute upper digit sum
        int upper_dsum = 0;
        for (int i = BLOCK; i < 20; i++) upper_dsum += digits[i];

        // Get bottom block value
        int bottom_val = 0;
        {
            int mul = 1;
            for (int i = 0; i < BLOCK; i++) {
                bottom_val += digits[i] * mul;
                mul *= 10;
            }
        }

        // Compute entry: how many steps until carry out of bottom block
        Entry e = get_entry(bottom_val, upper_dsum);

        if (steps_done + e.steps <= TARGET) {
            // Apply full block jump
            steps_done += e.steps;

            // Update bottom digits
            int nv = e.new_val;
            for (int i = 0; i < BLOCK; i++) {
                digits[i] = nv % 10;
                nv /= 10;
            }

            // Propagate carry into upper digits
            int carry = 1;
            for (int i = BLOCK; i < 20 && carry; i++) {
                digits[i] += carry;
                if (digits[i] >= 10) {
                    digits[i] -= 10;
                    carry = 1;
                } else {
                    carry = 0;
                }
            }
        } else {
            // Can't do full block jump. Do one step at a time.
            ll remaining = TARGET - steps_done;
            // Single step
            int ds = 0;
            for (int i = 0; i < 20; i++) ds += digits[i];
            // Add ds to the number
            int carry = ds;
            for (int i = 0; i < 20; i++) {
                digits[i] += carry;
                carry = digits[i] / 10;
                digits[i] %= 10;
                if (carry == 0) break;
            }
            steps_done++;
        }
    }

    // Print the result
    // Find highest non-zero digit
    int top = 19;
    while (top > 0 && digits[top] == 0) top--;

    // Print number
    for (int i = top; i >= 0; i--) cout << digits[i];
    cout << endl;

    return 0;
}