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
Keywords
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
Programs
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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
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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
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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 *)
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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
Comments