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A108421 Smallest number of ones needed to write in binary representation 2*n as sum of two primes. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 13 23:41 EDT 2021. Contains 343868 sequences. (Running on oeis4.)