A332341 Prime scale sequence (see comments).

-2, -3, 5, -7, -11, -13, 31, -17, -19, -23, 59, -29, -37, -41, 107, -43, -47, -53, -61, -67, 271, -71, -73, -79, 223, -83, -89, -97, 269, -101, -103, -109, 313, -113, -127, -131, -137, -139, 647, -149, -151, -157, 457, -163, -167, -173, 503, -179, -181, -191, -193, -197, 941

Offset

1,1

Comments

Take a double-pan balance scale and name the pans "negative" and "positive". At each step, the question is: "Is there an unused prime that would balance the scale if added to the positive pan?" If the answer is positive, add that prime to the positive pan. Otherwise, add the smallest unused prime to the negative pan.

Is the number of primes in the positive pan infinite?

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Example

2 and 3 unbalance the scale (and are negative), but 5 = 2 + 3 balances it (and is positive).

Mathematica

a[1]=-2; a[n_]:=a[n]=Module[{tab=Table[a[i], {i, 1, n-1}],
totalN=Abs[Total[Select[Table[a[i], {i, 1, n-1}], Negative]]],
totalP=Total[Select[Table[a[i], {i, 1, n-1}], Positive]],
l=NextPrime[Last[Select[Table[a[i], {i, 1, n-1}], Negative]], -1],
m=NextPrime[Abs[Last[Select[Table[a[i], {i, 1, n-1}], Negative]]]]},
If[totalN==totalP, If[PrimePi[tab[[-1]]]-PrimePi[Abs[tab[[-2]]]]==1, -NextPrime[tab[[-1]]],
If[FreeQ[Abs[tab], m], -m, While[!FreeQ[Abs[tab], m], m=NextPrime[m]]; -m]],
If[PrimeQ[totalN-totalP]&&FreeQ[Abs[tab], totalN-totalP], totalN-totalP,
If[FreeQ[Abs[tab], Abs[l]], l, While[!FreeQ[Abs[tab], Abs[l]], l=NextPrime[l, -1]]; l]]]]; a/@Range[53]

Prog

(Python)
from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    used, d, nextp = set(), 0, 2
    while True:
        if d > 0 and d not in used and isprime(d):
            used.add(d); yield d; d = 0
        while nextp in used:
            nextp = nextprime(nextp)
        used.add(nextp); yield -nextp; d += nextp
print(list(islice(agen(), 53))) # Michael S. Branicky, May 12 2022

Crossrefs

Cf. A000040, A101544, A332787, A332788.

Sequence in context: A342566 A067908 A236128 * A262283 A187614 A191077

Adjacent sequences: A332338 A332339 A332340 * A332342 A332343 A332344

Keyword

sign

Author

Ivan N. Ianakiev, Feb 10 2020

Status

approved