A237885 a(n) is the number of ways that 4n can be written as p+q (p>q) with p, q, (p-q)/2, 4n-(p-q)/2 all prime numbers.

0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1, 1, 0, 4, 0, 0, 2, 0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 5, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 3, 0, 0, 2, 0, 1, 3, 0, 0, 3, 1, 0, 3

Offset

1,12

Comments

2n=q+(p-q)/2; 6n=p+(4n-(p-q)/2).

Number of ways that 2*n can be written as a+b with a<b and a, b, a+2*b and 2*a+b all prime. - Robert Israel, Jun 07 2022

Links

Lei Zhou, Table of n, a(n) for n = 1..10000

Example

When n=4, 4n=16, 16=13+3, (13-3)/2=5, 16-5=11, all four numbers {3, 5, 11, 13} are prime numbers.  There is no other such four number set with this property, so a(4)=1;
When n=30, 4n=120.
  120=113+7, (113-7)/2=53, 120-53=67.  Set 1: {7, 53, 67, 113}.
  120=109+11, (109-11)/2=49=7*7, X
  120=107+13, (107-13)/2=47, 120-47=73. Set 2: {13, 47, 73, 107}.
  120=103+17, (103-17)/2=43, 120-43=77=7*11, X
  120=101+19, (101-19)/2=41, 120-41=79. Set 3: {19, 41, 79, 101}.
  120=97+23, (97-23)/2=37, 120-37=83. Set 4: {23, 37, 83, 97}.
  120=89+31, (89-31)/2=29, 120-29=91=7*13, X
  120=83+37, same with Set 4.
  120=79+41, same with Set 3.
  120=73+47, same with Set 2.
  120=67+53, same with Set 1.
  120=61+59, (61-59)/2=1, X
  So four acceptable sets have been found, and therefore a(30)=4.

Maple

N:= 100: # for a(1)..a(N)
V:= Vector(N):
P:= select(isprime, [seq(i, i=3..2*N, 2)]):
nP:= nops(P):
for i from 1 to nP do
  p:= P[i];
  for j from i+1 to nP do
    q:= P[j];
    if p+q > 2*N then break fi;
    r:= (p+q)/2;
    if isprime(p+2*q) and isprime(2*p+q) then
      V[r]:= V[r]+1
    fi
  od
od:
convert(V, list); # Robert Israel, Jun 08 2022

Mathematica

Table[qn = 4*n; p = 2*n - 1; ct = 0; While[p = NextPrime[p]; p < qn, q = qn - p; If[PrimeQ[q] && PrimeQ[(p - q)/2] && PrimeQ[qn - (p - q)/2], ct++]]; ct/2, {n, 1, 87}]4*n-1

Prog

(PARI) a(n)=my(s); forprime(p=2, n, if(isprime(2*n-p) && isprime(2*n+p) && isprime(4*n-p), s++)); s \\ Charles R Greathouse IV, Mar 15 2015

Crossrefs

Cf. A002375, A354834.

Sequence in context: A016424 A337985 A108913 * A341775 A139032 A182035

Adjacent sequences: A237882 A237883 A237884 * A237886 A237887 A237888

Keyword

nonn,easy

Author

Lei Zhou, Feb 14 2014

Status

approved