cp's OEIS Frontend

A357128 a(n) is the least even number k > 2 such that the sum of the lower elements and the sum of the upper elements in the Goldbach partitions of k are both divisible by 2^n, but not both divisible by 2^(n+1).

%I A357128 #14 Mar 23 2025 20:52:45
%S A357128 6,4,10,16,32,468,464,3576,14954,96000,403200
%N A357128 a(n) is the least even number k > 2 such that the sum of the lower elements and the sum of the upper elements in the Goldbach partitions of k are both divisible by 2^n, but not both divisible by 2^(n+1).
%C A357128 a(n) is the least even number k > 2 such that min(A007814(A185297(k/2)), A007814(A187129(k/2))) = n.
%e A357128 a(2) = 10 because the Goldbach partitions of 10 are 3+7 and 5+5, and 3+5 = 8 and 7+5 = 12 are both divisible by 2^2, but 12 is not divisible by 2^3; and 10 is the least even number > 2 that works.
%p A357128 N:= 10^4: # to use the first N primes
%p A357128 P:= [seq(ithprime(i),i=2..N)]:
%p A357128 M:= P[-1]+3:
%p A357128 L:= Vector(M): H:= Vector(M):
%p A357128 L[4]:= 2: H[4]:= 2:
%p A357128 for i from 1 to N-1 do
%p A357128   for j from i to N-1 do
%p A357128      t:= P[i]+P[j];
%p A357128      if t > M then break fi;
%p A357128      L[t]:= L[t]+P[i];
%p A357128      H[t]:= H[t]+P[j];
%p A357128 od od:
%p A357128 V:= Array(0..9): count:= 0:
%p A357128 for n from 4 by 2 to M while count < 10 do
%p A357128   v:= padic:-ordp(igcd(L[n],H[n]),2);
%p A357128   if V[v]=0 then count:= count+1; V[v]:= n; fi
%p A357128 od:
%p A357128 convert(V,list);
%Y A357128 Cf. A007814, A185297, A187129.
%K A357128 nonn,more
%O A357128 0,1
%A A357128 _J. M. Bergot_ and _Robert Israel_, Sep 13 2022