A332341 - OEIS

%I #21 May 12 2022 10:01:35

%S -2,-3,5,-7,-11,-13,31,-17,-19,-23,59,-29,-37,-41,107,-43,-47,-53,-61,

%T -67,271,-71,-73,-79,223,-83,-89,-97,269,-101,-103,-109,313,-113,-127,

%U -131,-137,-139,647,-149,-151,-157,457,-163,-167,-173,503,-179,-181,-191,-193,-197,941

%N Prime scale sequence (see comments).

%C 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.

%C Is the number of primes in the positive pan infinite?

%H Michael S. Branicky, <a href="/A332341/b332341.txt">Table of n, a(n) for n = 1..10000</a>

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

%t a[1]=-2;a[n_]:=a[n]=Module[{tab=Table[a[i],{i,1,n-1}],

%t totalN=Abs[Total[Select[Table[a[i],{i,1,n-1}],Negative]]],

%t totalP=Total[Select[Table[a[i],{i,1,n-1}],Positive]],

%t l=NextPrime[Last[Select[Table[a[i],{i,1,n-1}],Negative]],-1],

%t m=NextPrime[Abs[Last[Select[Table[a[i],{i,1,n-1}],Negative]]]]},

%t If[totalN==totalP,If[PrimePi[tab[[-1]]]-PrimePi[Abs[tab[[-2]]]]==1,-NextPrime[tab[[-1]]],

%t If[FreeQ[Abs[tab],m],-m,While[!FreeQ[Abs[tab],m],m=NextPrime[m]];-m]],

%t If[PrimeQ[totalN-totalP]&&FreeQ[Abs[tab],totalN-totalP],totalN-totalP,

%t If[FreeQ[Abs[tab],Abs[l]],l,While[!FreeQ[Abs[tab],Abs[l]],l=NextPrime[l,-1]];l]]]];a/@Range[53]

%o (Python)

%o from itertools import islice

%o from sympy import isprime, nextprime

%o def agen(): # generator of terms

%o     used, d, nextp = set(), 0, 2

%o     while True:

%o         if d > 0 and d not in used and isprime(d):

%o             used.add(d); yield d; d = 0

%o         while nextp in used:

%o             nextp = nextprime(nextp)

%o         used.add(nextp); yield -nextp; d += nextp

%o print(list(islice(agen(), 53))) # _Michael S. Branicky_, May 12 2022

%Y Cf. A000040, A101544, A332787, A332788.

%K sign

%O 1,1

%A _Ivan N. Ianakiev_, Feb 10 2020