A108421 Smallest number of ones needed to write in binary representation 2*n as sum of two primes.

2, 4, 4, 4, 5, 5, 5, 5, 4, 4, 5, 6, 5, 5, 6, 4, 5, 6, 5, 5, 5, 5, 6, 6, 6, 5, 6, 5, 6, 7, 7, 7, 8, 5, 5, 6, 5, 5, 6, 6, 5, 6, 6, 5, 6, 6, 7, 8, 5, 5, 6, 6, 6, 6, 7, 5, 6, 6, 7, 8, 7, 7, 8, 6, 7, 5, 5, 6, 5, 5, 6, 6, 5, 6, 6, 5, 6, 7, 7, 7, 6, 6, 6, 6, 6, 6, 7, 6, 6, 7, 7, 7, 8, 7, 8, 6, 5, 5, 6, 6, 6, 6, 7, 5, 6

Offset

2,1

Comments

a(n) = Min{A000120(p)+A000120(q) : p,q prime and p+q=2*n}.

a(n) = A108422(n) - A108423(n).

a(n) >= A000120(n)+1, with equality for n in A241757. - Robert Israel, Mar 25 2018

Links

Robert Israel, Table of n, a(n) for n = 2..10000

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to binary expansion of n

Example

n=15: 2*15=30 and A002375(15)=3 with 30=7+23=11+19=13+17,
13+17 -> 1101+10001 needs a(15)=5 binary ones, whereas
7+23 -> 111+10111 and 11+19 -> 1011+10011 need more.

Maple

N:= 200: # to get a(2)..a(N)
Primes:= select(isprime, [seq(i, i=3..2*N-3, 2)]):
Ones:= map(t -> convert(convert(t, base, 2), `+`), Primes):
V:= Vector(N): V[2]:= 2:
for i from 1 to nops(Primes) do
  p:= Primes[i];
  for j from 1 to i do
    k:= (p+Primes[j])/2;
    if k > N then break fi;
    t:= Ones[i]+Ones[j];
    if V[k] = 0 or t < V[k] then V[k]:= t fi
  od
od:
convert(V[2..N], list); # Robert Israel, Mar 25 2018

Mathematica

Min[#]&/@(Table[Total[Flatten[IntegerDigits[#, 2]]]&/@Select[ IntegerPartitions[ 2*n, {2}], AllTrue[#, PrimeQ]&], {n, 2, 110}]) (* Harvey P. Dale, Jul 27 2020 *)

Crossrefs

Cf. A000120, A004676, A005843, A007088, A108422, A108423, A241757.

Sequence in context: A162798 A088650 A097154 * A104058 A132345 A178976

Adjacent sequences: A108418 A108419 A108420 * A108422 A108423 A108424

Keyword

nonn,base

Author

Reinhard Zumkeller, Jun 03 2005

Status

approved