A332787 Negative-pan primes (see Comments).

2, 3, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 229, 233, 239, 241, 251, 257, 263, 277, 281, 283, 293, 307, 311, 317, 331, 337, 347, 349, 353, 359, 367

Offset: 1

Author

Ivan N. Ianakiev, Feb 24 2020

Keywords

• easy
• nonn

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.

The negative pan N can be fractalized, i.e., subdivided into NN and NP pans, where NN ={{2,3,7,11},{13,17,19,29,37,41,43},...} and NP = {{23},{199},...}. Can this fractalization be continued infinitely?

Examples

			First division: 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).
Second division: 2,3,7 and 11 unbalance the N pan (and go to the NN subpan), but 23 balances it (and goes to NP subpan).
		

Links

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

Crossrefs

Cf. A249031, A332341, A332788.

Programs

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]},
If[ totalN==totalP,
If[ PrimePi[tab[[-1]]]-PrimePi[Abs[tab[[-2]]]]==1,-NextPrime[tab[[-1]]],
NextPrime[tab[[-2]],-1]],
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]
]]];Abs[Select[a/@Range[78],Negative]]

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); d = 0
        while nextp in used:
            nextp = nextprime(nextp)
        used.add(nextp); yield nextp; d += nextp
print(list(islice(agen(), 65))) # Michael S. Branicky, May 12 2022