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

0,6

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

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

Prog

(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

Crossrefs

A288573 appears to be an erroneous version of this sequence.

Cf. A007963, A054860, A087916.

Sequence in context: A342929 A038669 A288573 * A376638 A135692 A089003

Adjacent sequences: A288571 A288572 A288573 * A288575 A288576 A288577

Keyword

nonn

Author

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

Status

approved