cp's OEIS Frontend

A262136 Number of distinct fractional parts of the numbers Sum_{i=j..k} (-1)^i/i with 1 <= j <= k <= n, where the fractional part of x is given by x - floor(x).

%I A262136 #29 Nov 16 2024 02:01:57
%S A262136 1,2,4,7,11,14,20,27,35,44,54,64,76,89,103,118,134,151,169,186,206,
%T A262136 227,249,272,296,321,347,374,402,430,460,491,523,556,590,625,661,698,
%U A262136 736,775,815,854,896,939,983,1028,1074,1121,1169,1218,1268,1319,1371,1424,1478,1532,1588,1645,1703,1762
%N A262136 Number of distinct fractional parts of the numbers Sum_{i=j..k} (-1)^i/i with 1 <= j <= k <= n, where the fractional part of x is given by x - floor(x).
%C A262136 Note that (-1)^n/n+(-1)^(n+1)/(n+1) = (-1)^n/(n*(n+1)) for any n > 0.
%C A262136 Conjecture: (i) Suppose that Sum_{i=j..k} (-1)^i/i and Sum_{r=s..t} (-1)^r/r with 0 < min{2,k} <= j <= k, 0 < min{2,t} <= s <= t and j <= s have the same fractional part, but the ordered pairs (j,k) and (s,t) are different. Then Sum_{i=j..k} (-1)^i/i = Sum_{r=s..t} (-1)^r/r. Moreover, if j is odd, then j > 1, k = j*(j+1) and (s,t) = (j+2,j*(j+1)-1); if j is even, then either (k = j+1 and s = t = j*(j+1)), or (k = j*(j+1)-1 and (s,t) = (j+2,j*(j+1))).
%C A262136 (ii) Let a > b >= 0 and m > 0 be integers with gcd(a,b) = 1 < max{a,m}. For each r = 0,1, the numbers Sum_{i=j..k} (-1)^(i-r*j)/(a*i-b)^m with 1 <= j <= k and (j > 1 if k > a-b = 1) have pairwise distinct fractional parts.
%C A262136 This is an analog of the conjecture in A261878. Part (i) of the conjecture implies that a(n) = n*(n-1)/2 + 2 - floor((sqrt(4n+1)-1)/2) - floor((sqrt(4n+1)-1)/4) for all n > 1.
%H A262136 Zhi-Wei Sun, <a href="/A262136/b262136.txt">Table of n, a(n) for n = 1..1000</a>
%e A262136 a(6) = 14 since the sums (-1)^j/j+...+(-1)^k/k with 0 < min{k,2} <= j <= k <= 6 and (j,k) different from (4,6) and (6,6) have pairwise distinct fractional parts, but (-1)^6/6 = (-1)^2/2+(-1)^3/3 and 1/4-1/5+1/6 = 1/2-1/3+1/4-1/5.
%t A262136 frac[x_]:=x-Floor[x]
%t A262136 u[0]:=0
%t A262136 u[n_]:=u[n-1]+(-1)^n/n
%t A262136 S[n_]:=Table[frac[u[n]-u[m-1]],{m,Min[2,n],n}]
%t A262136 T[1]:=S[1]
%t A262136 T[n_]:=Union[T[n-1],S[n]]
%t A262136 Do[Print[n," ",Length[T[n]]],{n,1,60}]
%Y A262136 Cf. A058312, A058313, A261878.
%K A262136 nonn
%O A262136 1,2
%A A262136 _Zhi-Wei Sun_, Sep 11 2015