A288574 Total number of distinct primes in all representations of 2*n+1 as a sum of 3 odd primes.

0, 0, 0, 0, 1, 2, 4, 4, 6, 7, 9, 10, 12, 15, 17, 16, 19, 19, 23, 25, 26, 26, 28, 33, 32, 35, 43, 39, 41, 45, 45, 48, 54, 55, 52, 60, 59, 56, 75, 67, 67, 81, 74, 76, 92, 83, 85, 100, 96, 81, 106, 103, 91, 121, 108, 98, 131, 120, 116, 143, 133, 129, 151, 144, 124, 163

Offset: 0

Author

Franklin T. Adams-Watters and R. J. Mathar, Jun 28 2017

Keywords

• nonn

Comments

That is, a representation 2n+1 = p+p+p counts as 1, as p+p+q counts as 2, and p+q+r counts as 3. If each representation is counted once, we simply get A007963.

Links

• Indranil Ghosh (first 200 terms), Hugo Pfoertner, Table of n, a(n) for n = 0..10000

Crossrefs

A288573 appears to be an erroneous version of this sequence.

Cf. A007963, A054860, A087916.

Programs

Maple

A288574 := proc(n)
    local a, i, j, k, p, q, r,pqr ;
    a := 0 ;
    for i from 2 do
        p := ithprime(i) ;
        for j from i do
            q := ithprime(j) ;
            for k from j do
                r := ithprime(k) ;
                if p+q+r = 2*n+1 then
                    pqr := {p,q,r} ;
                    a := a+nops(pqr) ;
                elif p+q+r > 2*n+1 then
                    break;
                end if;
            end do:
            if p+2*q > 2*n+1 then
                break;
            end if;
        end do:
        if 3*p > 2*n+1 then
            break;
        end if;
    end do:
    return a;
end proc:
seq(A288574(n),n=0..80) ; # R. J. Mathar, Jun 29 2017

Mathematica

Table[x = 2 n + 1; max = PrimePi[x]; Total[Length /@ Tally /@ DeleteDuplicates[Sort /@ Select[Tuples[Range[2, max], 3], Prime[#[[1]]] + Prime[#[[2]]] + Prime[#[[3]]] == x &]]], {n, 0, 100}] (* Robert Price, Apr 22 2025 *)

PARI

a(n)={my(p,q,r,cnt);n=2*n+1;
forprime(p=3,n\3,forprime(q=p,(n-p)\2,
if(isprime(r=n-p-q), cnt+=if(p===q&&p==r,1,if(p==q||q==r,2,3)))));cnt}
\\ Franklin T. Adams-Watters, Jun 28 2017

Python

from sympy import primerange, isprime
def a(n):
    n=2*n + 1
    c=0
    for p in primerange(3, n//3 + 1):
        for q in primerange(p, (n - p)//2 + 1):
            r=n - p - q
            if isprime(r): c+=1 if p==q and p==r else 2 if p==q or q==r else 3
    return c
print([a(n) for n in range(66)]) # Indranil Ghosh, Jun 29 2017