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

Author

N. J. A. Sloane, Sep 29 2019

Keywords

• nonn

Comments

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.

Examples

			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)
		

References

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

Links

• 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)

Crossrefs

Cf. A007504, A140358.

Programs

Mathematica

(* 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 *)

Python

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

Formula

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

Extensions

More terms from Giovanni Resta, Sep 30 2019