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

%I A327467 #37 May 30 2023 07:43:55
%S A327467 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,
%T A327467 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,
%U A327467 9,10,9,8,9,8,9,10,9,8,9,10,9,10,9,10,9,10
%N A327467 a(n) = smallest k such that n can be expressed as a signed sum of the first k primes.
%C A327467 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.
%D A327467 Allan C. Wechsler, Posting to Sequence Fans Mailing List, circa Aug 29 2019.
%H A327467 Seiichi Manyama, <a href="/A327467/b327467.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Giovanni Resta)
%H A327467 Karl-Heinz Hofmann, <a href="/A327467/a327467.txt">Examples for n = 0 to 778</a>
%H A327467 Karl-Heinz Hofmann, <a href="/A327467/a327467.png">Visualization of the conjecture of _Kei Fujimoto_ (see formula)</a>
%F A327467 a(A007504(n)) = n for n > 0. - _Seiichi Manyama_, Sep 30 2019
%F A327467 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
%e A327467 Illustration of initial terms:
%e A327467 0  =   2 + 3 - 5
%e A327467 1  = - 2 + 3
%e A327467 2  =   2
%e A327467 3  = - 2 + 3 - 5 + 7
%e A327467 4  =   2 - 3 + 5
%e A327467 5  =   2 + 3
%e A327467 6  = - 2 + 3 + 5
%e A327467 7  =   2 + 3 - 5 + 7
%e A327467 8  =   2 - 3 + 5 - 7 + 11
%e A327467 9  =   2 - 3 + 5 + 7 + 11 - 13
%e A327467 10 =   2 + 3 + 5
%e A327467 (for more examples see links)
%t A327467 (* 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 *)
%o A327467 (Python)
%o A327467 from sympy import sieve as prime
%o A327467 def A327467(n):
%o A327467     array, np, k = [2], 1, 1
%o A327467     while n not in array:
%o A327467         temp = []; np += 1; k += 1
%o A327467         for item in array:
%o A327467             temp.append(item + prime[k])
%o A327467             temp.append(abs(item - prime[k]))
%o A327467         array = set(temp)
%o A327467     return np
%o A327467 print([A327467(n) for n in range(0, 100)]) # _Karl-Heinz Hofmann_, May 30 2023
%Y A327467 Cf. A007504, A140358.
%K A327467 nonn
%O A327467 0,1
%A A327467 _N. J. A. Sloane_, Sep 29 2019
%E A327467 More terms from _Giovanni Resta_, Sep 30 2019