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

Author

Lei Zhou, Feb 14 2014

Keywords

• nonn
• easy

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 aRobert Israel, Jun 07 2022

Examples

			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.
		

Links

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

Crossrefs

Cf. A002375, A354834.

Programs

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

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