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

1,1

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

Cf. A075326, A332341, A332787.

Sequence in context: A245523 A147033 A125743 * A333243 A078686 A031908

Adjacent sequences: A332785 A332786 A332787 * A332789 A332790 A332791

Keyword

nonn

Author

Ivan N. Ianakiev, Feb 24 2020

Extensions

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

Status

approved