A140358 -id:A140358 - cp's OEIS frontend

A327467 a(n) = smallest k such that n can be expressed as a signed sum of the first k primes.

3, 2, 1, 4, 3, 2, 3, 4, 5, 6, 3, 4, 5, 4, 5, 6, 7, 4, 5, 6, 7, 6, 5, 6, 5, 6, 7, 6, 5, 8, 7, 6, 7, 8, 7, 6, 7, 6, 7, 8, 7, 6, 7, 8, 7, 8, 9, 8, 7, 8, 9, 8, 7, 8, 7, 8, 9, 8, 7, 8, 9, 8, 9, 8, 9, 10, 9, 8, 9, 10, 9, 8, 9, 8, 9, 10, 9, 8, 9, 10, 9, 10, 9, 10, 9, 10

Offset: 0

N. J. A. Sloane, Sep 29 2019

nonn

Smallest k such that n = +- p_1 +- p_2 +- p_3 +- ... +- p_k for a suitable choice of signs, where p_i = i-th prime.

			Illustration of initial terms:
0  =   2 + 3 - 5
1  = - 2 + 3
2  =   2
3  = - 2 + 3 - 5 + 7
4  =   2 - 3 + 5
5  =   2 + 3
6  = - 2 + 3 + 5
7  =   2 + 3 - 5 + 7
8  =   2 - 3 + 5 - 7 + 11
9  =   2 - 3 + 5 + 7 + 11 - 13
10 =   2 + 3 + 5
(for more examples see links)

Allan C. Wechsler, Posting to Sequence Fans Mailing List, circa Aug 29 2019.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Giovanni Resta)
Karl-Heinz Hofmann, Examples for n = 0 to 778
Karl-Heinz Hofmann, Visualization of the conjecture of _Kei Fujimoto_ (see formula)

Cf. A007504, A140358.

(* 1001 terms *) sgn[w_] := Union@ Abs[Total /@ (w # & /@ Tuples[{1, -1}, Length@w])]; set[n_] := Block[{h = Floor[n/2], p = Prime@ Range@ n, x, y}, x = sgn[Take[p, h]]; y = sgn[Take[p, h - n]]; Union@ Flatten@ Table[{e + f, Abs[e - f]}, {e, x}, {f, y}]]; T = {}; L = 0 Range[1001]; k = 0; While[Length[T] < 1001, k++; s = Select[set[k], # <= 1000 && ! MemberQ[T, #] &]; Do[L[[e + 1]] = k, {e, s}]; T = Union[T, s]]; L (* Giovanni Resta, Sep 30 2019 *)

from sympy import sieve as prime
def A327467(n):
    array, np, k = [2], 1, 1
    while n not in array:
        temp = []; np += 1; k += 1
        for item in array:
            temp.append(item + prime[k])
            temp.append(abs(item - prime[k]))
        array = set(temp)
    return np
print([A327467(n) for n in range(0, 100)]) # Karl-Heinz Hofmann, May 30 2023

a(A007504(n)) = n for n > 0. - Seiichi Manyama, Sep 30 2019

Conjecture. Let k be the smallest integer satisfying n<=A007504(k). If n=9 or 16, a(n)=k+3 (so a(9)=6, a(16)=7), else if A007504(k)-n is odd, a(n)=k+1. If A007504(k)-n=2 or 8 or 12, a(n)=k+2, otherwise a(n)=k. - Kei Fujimoto, Sep 24 2021

More terms from Giovanni Resta, Sep 30 2019

A197702 Smallest positive integer k such that n = +-1 +-3 +-... +-(2k-1) for some choice of +'s and -'s.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 3, 4, 3, 4, 5, 6, 5, 4, 5, 4, 5, 6, 5, 6, 7, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 8, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 10, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 11, 10, 9, 10, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 12, 11, 10, 11, 10

Offset: 1

John W. Layman, Oct 18 2011

nonn

Conjecture. Let SO(k) be the sum of the first k odd positive integers. Then a(n)=k if n=SO(k). Otherwise, choose k so that SO(k-1)

			The sum of 3 terms 1 - 3 + 5 gives 3, but none of the 2-term sums 1+3, 1-3, -1+3, -1-3 gives 3, so a(3)=3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A140358.

b:= proc(n, i) option remember; (n=0 and i=0) or
      abs(n)<=i^2 and (b(n-2*i+1, i-1) or b(n+2*i-1, i-1))
    end:
a:= proc(n) local k;
      for k from floor(sqrt(n)) while not b(n, k) do od; k
    end:
seq(a(n), n=1..100); # Alois P. Heinz, Oct 19 2011

b[n_, i_] := b[n, i] = (n==0 && i==0) || Abs[n] <= i^2 && (b[n-2i+1, i-1] || b[n+2i-1, i-1]);
a[n_] := Module[{k}, For[k = Floor[Sqrt[n]], !b[n, k], k++]; k];
Array[a, 100] (* Jean-François Alcover, Nov 12 2020, after Alois P. Heinz *)

Showing 1-2 of 2 results.

cp's OEIS Frontend

A327467 a(n) = smallest k such that n can be expressed as a signed sum of the first k primes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions