OFFSET
0,1
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.
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)
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
EXAMPLE
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)
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 29 2019
EXTENSIONS
More terms from Giovanni Resta, Sep 30 2019
STATUS
approved