A332788 Positive-pan primes (see Comments).

5, 31, 59, 107, 271, 223, 269, 313, 647, 457, 503, 941, 1579, 12919, 3571, 1667, 2897, 3037, 3187, 1993, 3461, 2179, 10141, 5927, 11969, 4957, 13627, 5519, 22787, 3851, 3889, 3929, 15217, 44221, 65867, 21799, 10211, 31727, 24623, 11467, 6983, 7039, 16651, 73351

Offset: 1

Author

Ivan N. Ianakiev, Feb 24 2020

Keywords

• 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.

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)?

Examples

			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).
		

Links

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

Crossrefs

Cf. A075326, A332341, A332787.

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],
    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]

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

Extensions

a(41) and beyond from Michael S. Branicky, May 12 2022