PROBLEM. For k>=1 let a(k) be the least N such that there exist k DISTINCT integers x_1..x_k in {1..N}, listed in SOME order (any order, not necessarily increasing), such that all consecutive-block sums x_u+...+x_v (1<=u<=v<=k) are pairwise distinct (as numbers, across all lengths). Known: (3/2-o(1))k <= a(k) <= (3+o(1))k. CLAIM: a(k) > (3/2+1e-10)k for all k >= 1e22.

PROOF (eps=1e-10, eta=1e-4, K=1e22). Suppose k>=K, admissible row with N<=(3/2+eps)k. S=sum x_i, s_0=0, s_j=x_1+..+x_j.
1. For h<=k there are m=hk-h(h-1)/2 blocks of length<=h; their sums are distinct positive integers so total>=m(m+1)/2. Each entry lies in <=T=h(h+1)/2 such blocks, so S>=m(m+1)/(h(h+1)). With h=floor(sqrt k): S>=k^2-2k^{3/2}+k, so (1) S>=k^2-2k^{3/2}.
2. Distinctness gives S<=Smax=kN-k(k-1)/2<=(1+eps)k^2+k/2. E=Smax-S<=eps k^2+2k^{3/2}+k/2<=2eps k^2. Sort y_1<..<y_k; B=N-k+1; y_j<=B+j-1 and E=sum(B+j-1-y_j), all terms>=0. (1) with S<=Smax gives B>=k/2-2sqrt k+1/2. L=ceil((1/2-eta)k); B-L>=eta k-2sqrt k-1/2>=eta k/2. 'Bad' entry: value<L; q=#bad; each contributes >=B-L to E, so q<=2E/(eta k)<=4eps k/eta=k/250000.
3. Fix 1<=h<=8, B_h = set of block sums of lengths<=h, T=h(h+1)/2, m=hk-h(h-1)/2. Discard blocks containing a bad entry (<=Tq of them); M remain; d=hk-M<=h(h-1)/2+Tq<=28+36q, d/k<=1/6000. Retained sums z_1<..<z_M are distinct integers each >=L, so z_j>=L+j-1. Delta=sum(z_j-(L+j-1))>=0. Total of all block sums of length<=h is <=TS, so Delta<=TS-ML-M(M-1)/2 <= (T eps+h eta)k^2+(h+1/2-eta)dk-d^2/2+(T+h)k/2-d/2 <= (T eps+h eta)k^2+(h+1/2)dk+(T+h)k/2. Hence Delta/k^2<=36eps+8eta+17/12000+22/k<=9/4000.
4. LEMMA. F_h(f)=(1/k)sum_{b in B_h} f(b/k), F_0=0. For |f|<=1, Lip(f)<=10, 0<=h<=8: |F_h(f)-int_{1/2}^{h+1/2} f| <= 1/40. Errors: discarding <= d/k; moving z_j to L+j-1 <= 10Delta/k^2; Riemann sum over [L/k,(L+M)/k] <= 10M/k^2<=80/k; changing interval to [1/2,h+1/2] <= 2|L/k-1/2|+|M/k-h| <= 2eta+2/k+d/k. Total <= 90/4000+2/6000+2/10000+82/k <= 1/40.
5. For r=1..8, C_r=sum_{i=0}^{k-r} cos(3pi((s_{i+r}-s_i)/k - r)). f_r(t)=cos(3pi(t-r)), |f_r|<=1, Lip 3pi<10. B_r minus B_{r-1} is exactly the length-r block sums, so C_r/k=F_r(f_r)-F_{r-1}(f_r) <= int_{r-1/2}^{r+1/2} cos(3pi(t-r))dt + 1/20 = -2/(3pi)+1/20 <= -3/20.
6. u_j=exp(3pi i (s_j/k - j)), 0<=j<=k. Q=sum_{i=0}^{k-8} |sum_{j=0}^{8} u_{i+j}|^2 >= 0. Diagonal 9(k-7). For lag r, 9-r offsets per window; each offset's correlation sum omits exactly 8-r terms of C_r, each omitted cos>=-1, so partial<=C_r+(8-r). So 0<=Q<=9(k-7)+2sum_{r=1}^8 (9-r)C_r+2sum(9-r)(8-r)=9k+2sum(9-r)C_r+273 <= 9k+72(-3k/20)+273=-9k/5+273<0. Contradiction.