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 yes, 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 (P) infinite? If it is, is fractalization of P, i.e., further subdivision into PN and PP subpans, possible (including infinitely)?
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..10000
EXAMPLE
2 and 3 unbalance the scale (and go to the negative pan N), but 5 = 2 + 3 balances it (and goes to the positive pan P).
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]
]]]; Select[a/@Range[370], Positive]
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); d += nextp
print(list(islice(agen(), 44))) # Michael S. Branicky, May 12 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Ivan N. Ianakiev, Feb 24 2020
EXTENSIONS
a(41) and beyond from Michael S. Branicky, May 12 2022
STATUS
approved