#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #define rep(i,n) for(int (i)=0;(i)<(int)(n);++(i)) #define rer(i,l,u) for(int (i)=(int)(l);(i)<=(int)(u);++(i)) #define reu(i,l,u) for(int (i)=(int)(l);(i)<(int)(u);++(i)) #if defined(_MSC_VER) || __cplusplus > 199711L #define aut(r,v) auto r = (v) #else #define aut(r,v) __typeof(v) r = (v) #endif #define each(it,o) for(aut(it, (o).begin()); it != (o).end(); ++ it) #define all(o) (o).begin(), (o).end() #define pb(x) push_back(x) #define mp(x,y) make_pair((x),(y)) #define mset(m,v) memset(m,v,sizeof(m)) #define INF 0x3f3f3f3f #define INFL 0x3f3f3f3f3f3f3f3fLL using namespace std; typedef vector vi; typedef pair pii; typedef vector > vpii; typedef long long ll; template inline void amin(T &x, U y) { if(y < x) x = y; } template inline void amax(T &x, U y) { if(x < y) x = y; } template struct ModInt { static const int Mod = MOD; unsigned x; ModInt(): x(0) { } ModInt(signed sig) { int sigt = sig % MOD; if(sigt < 0) sigt += MOD; x = sigt; } ModInt(signed long long sig) { int sigt = sig % MOD; if(sigt < 0) sigt += MOD; x = sigt; } int get() const { return (int)x; } ModInt &operator+=(ModInt that) { if((x += that.x) >= MOD) x -= MOD; return *this; } ModInt &operator-=(ModInt that) { if((x += MOD - that.x) >= MOD) x -= MOD; return *this; } ModInt &operator*=(ModInt that) { x = (unsigned long long)x * that.x % MOD; return *this; } ModInt &operator/=(ModInt that) { return *this *= that.inverse(); } ModInt operator+(ModInt that) const { return ModInt(*this) += that; } ModInt operator-(ModInt that) const { return ModInt(*this) -= that; } ModInt operator*(ModInt that) const { return ModInt(*this) *= that; } ModInt operator/(ModInt that) const { return ModInt(*this) /= that; } ModInt inverse() const { long long a = x, b = MOD, u = 1, v = 0; while(b) { long long t = a / b; a -= t * b; std::swap(a, b); u -= t * v; std::swap(u, v); } return ModInt(u); } }; typedef ModInt<1000000007> mint; vector fact, factinv; void nCr_computeFactinv(int N) { N = min(N, mint::Mod - 1); fact.resize(N+1); factinv.resize(N+1); fact[0] = 1; rer(i, 1, N) fact[i] = fact[i-1] * i; factinv[N] = fact[N].inverse(); for(int i = N; i >= 1; i --) factinv[i-1] = factinv[i] * i; } vector isprime; vector primes; void sieve(int n){ if((int)isprime.size() >= n+1) return; isprime.assign(n+1, true); isprime[0] = isprime[1] = false; int sqrtn = (int)(sqrt(n * 1.) + .5); for(int i = 2; i <= sqrtn; i ++) if(isprime[i]) { for(int j = i * i; j <= n; j += i) isprime[j] = false; } primes.clear(); for(int i = 2; i <= n; i ++) if(isprime[i]) primes.push_back(i); } vector mobiusMu; void calcMobiusMu() { int n = (int)isprime.size() - 1; mobiusMu.assign(n+1, 1); for(int i = 2; i <= n; i ++) if(isprime[i]) { if((ll)i * i <= n) { for(int j = i * i; j <= n; j += i * i) mobiusMu[j] = 0; } for(int j = i; j <= n; j += i) mobiusMu[j] *= -1; } } typedef int FactorsInt; typedef vector > Factors; void primeFactors(FactorsInt x, Factors &out_v) { out_v.clear(); int sqrtx = (int)(sqrt(x*1.) + 10.5); sieve(sqrtx); for(vector::const_iterator p = primes.begin(); p != primes.end(); ++ p) { if(*p > sqrtx) break; if(x % *p == 0) { int t = 1; x /= *p; while(x % *p == 0) { t ++; x /= *p; } out_v.push_back(make_pair(*p, t)); } } if(x != 1) out_v.push_back(make_pair(x, 1)); } void getDivisors(FactorsInt x, vector &out_v) { Factors fs; primeFactors(x, fs); out_v.assign(1, 1); rep(i, fs.size()) { for(int j = (int)out_v.size()-1; j >= 0; j --) { FactorsInt x = out_v[j]; rep(k, fs[i].second) { x *= fs[i].first; out_v.push_back(x); } } } sort(all(out_v)); } vector allv; void brute(vi &C, vi &v) { bool emp = true; rep(i, C.size()) if(C[i] > 0) { emp = false; -- C[i]; v.push_back(i); brute(C, v); v.pop_back(); ++ C[i]; } if(emp) allv.pb(v); } int main() { int K; scanf("%d", &K); vector C(K); rep(i, K) scanf("%d", &C[i]); { allv.clear(); vi v; brute(C,v); } int N = accumulate(all(C), 0); nCr_computeFactinv(N); sieve(N); calcMobiusMu(); vector cnts(N+1); rer(p, 1, N) if(N % p == 0) { bool ok = true; rep(i, K) ok &= C[i] % (N / p) == 0; if(!ok) continue; mint y = fact[p]; rep(i, K) y *= factinv[C[i] / (N / p)]; cnts[p] = y; } mint ans = 0; rer(p, 1, N) if(N % p == 0) { //ちょうどp個の回転で同じになるようなものを数える //= 最小periodがちょうどp //periodがpの約数であるのは数えられるので包除原理したい bool ok = true; rep(i, K) ok &= C[i] % (N / p) == 0; if(!ok) continue; vector divs; getDivisors(p, divs); mint x = 0; each(d, divs) x += cnts[*d] * mobiusMu[p / *d]; ans += x * (N / p); } ans /= N; printf("%d\n", ans.get()); return 0; }