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).

6, 4, 10, 16, 32, 468, 464, 3576, 14954, 96000, 403200

Offset: 0

Author

J. M. Bergot and Robert Israel, Sep 13 2022

Keywords

• nonn
• more

Comments

a(n) is the least even number k > 2 such that min(A007814(A185297(k/2)), A007814(A187129(k/2))) = n.

Examples

			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.
		

Crossrefs

Cf. A007814, A185297, A187129.

Programs

Maple

N:= 10^4: # to use the first N primes
P:= [seq(ithprime(i),i=2..N)]:
M:= P[-1]+3:
L:= Vector(M): H:= Vector(M):
L[4]:= 2: H[4]:= 2:
for i from 1 to N-1 do
  for j from i to N-1 do
     t:= P[i]+P[j];
     if t > M then break fi;
     L[t]:= L[t]+P[i];
     H[t]:= H[t]+P[j];
od od:
V:= Array(0..9): count:= 0:
for n from 4 by 2 to M while count < 10 do
  v:= padic:-ordp(igcd(L[n],H[n]),2);
  if V[v]=0 then count:= count+1; V[v]:= n; fi
od:
convert(V,list);