A108423 -id:A108423 - cp's OEIS frontend

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

Reinhard Zumkeller, Jun 03 2005

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

			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.

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

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

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

A108422 Greatest number of ones that can be used to write in binary representation 2*n as sum of two primes.

Original entry on oeis.org

2, 4, 4, 5, 5, 6, 5, 6, 6, 6, 6, 6, 6, 7, 6, 7, 7, 8, 7, 8, 8, 8, 7, 8, 8, 9, 7, 8, 9, 10, 7, 8, 8, 9, 8, 9, 9, 10, 8, 9, 9, 8, 9, 10, 10, 10, 8, 8, 9, 9, 9, 10, 10, 10, 9, 9, 8, 10, 10, 10, 8, 10, 8, 9, 9, 10, 9, 10, 10, 9, 9, 10, 9, 11, 9, 10, 11, 12, 9, 10, 10, 10, 10, 11, 9, 12, 8, 9, 11, 10, 8, 12

Offset: 2

Reinhard Zumkeller, Jun 03 2005

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

a(n) = A108423(n) + A108421(n).

Cf. A004676, A005843, A007088, A108421.

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 t > V[k] then V[k]:= t fi
  od
od:
convert(V[2..N], list); # Robert Israel, Mar 26 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica