Let x be an infinite sequence whose entries are all +1 or -1. For this problem it is convenient to use 1-origin indexing, so the first element of x is x[1]. Now contruct the sequence d1 of partial sums: d1[0] = 0, d1[i+1] = d1[i] + x[i+1]. A similar sequence d2 adds only the even entries of x: d2[0] = 0, d2[2i+1] = d2[2i], d2[2i+2] = d2[2i] + x[2i+2] In general, we define dk[i] to equal the sum of all the elements x[j] such that j is a multiple of k not exceeding i. The "Erdos discrepancy" E[i] is the maximum over all k not exceeding i of | dk[i] |. Erdos conjectured that for all sequences x, E[i] eventually exceeds any given limit. The conjecture is still open, but generally believed to be true. Search the Web for "polymath5" (one word) for many references. You can quickly prove that E[i] exceeds 1 for some i < 12. For a limit of 2, things are not as clear. We have a many sequences of length 1124 that manage to keep below E[i]=3, but none of length 1125. It has been proved that all sequences reach E[i]=3 before somewhere between 2000 and 2100 (I forget the best known upper bound). OEIS's A181740 tabulates how many sequences of length n manage to keep E[i] <= 2. It gives values up to A[29], provided by Jeffrey Shallit. The current state of belief is that A[N] is 0 for all N exceeding 1124. I just wrote a program to calculate some more values; it agrees with Shallit's results as far as they go. I am planning to enter my new values (so far, up to A[50]), but I'm hoping that somebody can replicate my results. I list them below. A[30 .. 39] : 81326, 122422, 244844, 234934, 356154, 309068, 388042, 589796, 900000 [sic!], 813466 A[40 .. 49] : 1212450, 1837030, 1882194, 2921946, 4544342, 2274560, 3542738, 5495686, 8436986, 9597362 A[50] : 11352364 My program is really dumb; perhaps somebody can come up with a faster version. Mine is in Java; just rewriting it in C will probably speed it up enormously.