A332788 -id:A332788 - cp's OEIS frontend

A332341 Prime scale sequence (see comments).

-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

Ivan N. Ianakiev, Feb 10 2020

sign

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?

			2 and 3 unbalance the scale (and are negative), but 5 = 2 + 3 balances it (and is positive).

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

Cf. A000040, A101544, A332787, A332788.

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]

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

A332787 Negative-pan primes (see Comments).

Original entry on oeis.org

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

Ivan N. Ianakiev, Feb 24 2020

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?

			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).

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

Cf. A249031, A332341, A332788.

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]]

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

cp's OEIS Frontend

A332341 Prime scale sequence (see comments).

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