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A332341
Prime scale sequence (see comments).
3
-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
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
KEYWORD
sign
AUTHOR
Ivan N. Ianakiev, Feb 10 2020
STATUS
approved