A071704 Number of ways to represent the n-th prime as arithmetic mean of three other odd primes.

0, 0, 0, 2, 5, 7, 10, 14, 16, 24, 29, 31, 42, 40, 43, 52, 62, 70, 75, 87, 82, 96, 102, 112, 127, 137, 136, 142, 154, 154, 186, 199, 204, 215, 233, 248, 250, 262, 272, 284, 309, 324, 344, 334, 348, 358, 406, 414, 430, 446, 441, 489, 486, 511, 508

Offset: 1

Author

Reinhard Zumkeller, Jun 03 2002

Keywords

• nonn

Examples

			a(5)=5 as A000040(5)=11 and there are no more representations not containing 11 than 11 = (3+7+23)/3 = (3+13+17)/3 = (5+5+23)/3 = (7+7+19)/3 = (7+13+13)/3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A071681, A071703, A000040, A065091, A258233.

Programs

Haskell

a071704 n = z (us ++ vs) 0 (3 * q)  where
   z _ 3 m = fromEnum (m == 0)
   z ps'@(p:ps) i m = if m < p then 0 else z ps' (i+1) (m - p) + z ps i m
   (us, _:vs) = span (< q) a065091_list; q = a000040 n
-- Reinhard Zumkeller, May 24 2015

Maple

N:= 300: # to get the first A000720(N) terms
P:= select(isprime, [seq(i,i=3..3*N,2)]):
nP:= nops(P):
V:= Vector(N):
for i from 1 to nP do
  for j from i to nP do
    for k from j to nP while P[i]+P[j]+P[k] <= 3*N do
      r:= (P[i]+P[j]+P[k])/3;
      if r::integer and isprime(r) and r <> P[j] and r <= N then V[r]:= V[r]+1 fi
od od od:
seq(V[ithprime(i)],i=1..numtheory:-pi(N)); # Robert Israel, Aug 09 2018

Mathematica

M = 300; (* to get the first A000720(M) *)
P = Select[Range[3, 3*M, 2], PrimeQ]; nP = Length[P]; V = Table[0, {M}];
For[i = 1, i <= nP, i++,
For[j = i, j <= nP, j++,
For[k = j, k <= nP && P[[i]] + P[[j]] + P[[k]] <= 3*M , k++, r = (P[[i]] + P[[j]] + P[[k]])/3; If[IntegerQ[r] && PrimeQ[r] && r != P[[j]] && r <= M, V[[r]] = V[[r]]+1]
]]];
Table[V[[Prime[i]]], {i, 1, PrimePi[M]}] (* Jean-François Alcover, Mar 09 2019, after Robert Israel *)

PARI

a(n, p=prime(n))=my(s=0); forprime(q=p+2, 3*p-4, my(t=3*p-q); forprime(r=max(t-q, 3), (3*p-q)\2, if(t!=p+r && isprime(t-r), s++))); s \\ Charles R Greathouse IV, Jun 04 2015

Extensions

Definition corrected by Zak Seidov, May 24 2015