A232111 -id:A232111 - cp's OEIS frontend

A232112 Denominator of smallest nonnegative fraction of form +- 1 +- 1/2 +- 1/3 ... +- 1/n.

1, 1, 2, 6, 12, 60, 20, 420, 840, 2520, 2520, 27720, 27720, 360360, 360360, 360360, 144144, 12252240, 12252240, 232792560, 232792560, 232792560, 46558512, 5354228880, 5354228880, 2974571600, 26771144400, 80313433200

Offset: 0

David W. Wilson, Nov 18 2013

Numerators are A232111.

			1-1/2-1/3-1/4+1/5 = 7/60. No other choice of term signs yields a smaller nonnegative fraction, so a(5) = 60.
0/1, 1/1, 1/2, 1/6, 1/12, 7/60, 1/20, 11/420, 13/840, 11/2520, 11/2520, 23/27720, 23/27720, 607/360360, 251/360360, 251/360360, 25/144144, 97/12252240, ...

David W. Wilson, Table of n, a(n) for n = 0..40

Cf. A232111.

nMax = 19; d = {0}; Table[d = Flatten[{d + 1/n, d - 1/n}]; Denominator[Min[Abs[d]]], {n, nMax}] (* T. D. Noe, Nov 20 2013 *)

a(n,t=0)=if(n==1,denominator(abs(n-t)),min(a(n-1,t-1/n),a(n-1,t+1/n))) \\ Charles R Greathouse IV, Apr 06 2014

from math import lcm, gcd
from itertools import product
def A232112(n):
    if n <= 1: return 1
    m = lcm(*range(2,n+1))
    mtuple = tuple(m//i for i in range(2,n+1))
    return m//gcd(m,min(abs(m+sum(d[i]*mtuple[i] for i in range(n-1))) for d in product((-1,1),repeat=n-1))) # Chai Wah Wu, Nov 25 2021

A349544 Smallest possible value of |Sum_{k=0..n} (+-) 2^k * 3^(n-k)|, where each (+-) can be either plus or minus sign, independently for each term in the sum.

Original entry on oeis.org

1, 1, 1, 5, 1, 19, 7, 5, 65, 61, 73, 227, 257, 5, 439, 1253, 2425, 2035, 833, 2677, 10591, 6509, 32071, 41173, 77263, 114323, 18145, 129685, 321151, 15757, 645449, 113957, 50735, 477653, 24295, 5089013, 3743881, 4809115, 12209455, 8216179, 32894927, 80299843, 45673913

Offset: 0

Vladimir Reshetnikov, Nov 21 2021

All terms are positive odd integers.

			For n = 3, there are 2^3 = 8 possible choices of signs: 3^3 + 2*3^2 + 2^2*3 + 2^3 = 65, 3^3 + 2*3^2 + 2^2*3 - 2^3 = 49, 3^3 + 2*3^2 - 2^2*3 + 2^3 = 41, 3^3 + 2*3^2 - 2^2*3 - 2^3 = 25, 3^3 - 2*3^2 + 2^2*3 + 2^3 = 29, 3^3 - 2*3^2 + 2^2*3 - 2^3 = 13, 3^3 - 2*3^2 - 2^2*3 + 2^3 = 5, and 3^3 - 2*3^2 - 2^2*3 - 2^3 = -11. The smallest absolute value is 5, so a(3) = 5.

Math StackExchange, Minimizing a generalized partial sum of a geometric progression

Cf. A001047, A036561, A232111, A232112.

b:= proc(k, n) option remember; `if`(k<0, {0}, map(x->
     (t-> [x+t, abs(x-t)][])(2^(n-k)*3^k), b(k-1, n)))
    end:
a:= n-> min(b(n$2)):
seq(a(n), n=0..18);  # Alois P. Heinz, Nov 21 2021

Min@*Abs/@FoldList[Join[3 #1 + 2^#2, 3 #1 - 2^#2] &, {1}, Range[25]]

def f(k,n):
    if k == 0 and n == 0: return (x for x in (1,))
    if k < n: return (y*3 for y in f(k,n-1))
    return (abs(x+y) for x in f(k-1,n) for y in (2**n,-2**n))
def A349544(n): return min(f(n,n)) # Chai Wah Wu, Nov 24 2021

a(33)-a(35) from Chai Wah Wu, Nov 24 2021

a(36)-a(42) from Martin Ehrenstein, Nov 26 2021

cp's OEIS Frontend

A232112 Denominator of smallest nonnegative fraction of form +- 1 +- 1/2 +- 1/3 ... +- 1/n.

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