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%I #41 May 08 2021 23:06:29
%S 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,
%T 43,39,41,45,45,48,54,55,52,60,59,56,75,67,67,81,74,76,92,83,85,100,
%U 96,81,106,103,91,121,108,98,131,120,116,143,133,129,151,144,124,163
%N Total number of distinct primes in all representations of 2*n+1 as a sum of 3 odd primes.
%C 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.
%H Indranil Ghosh (first 200 terms), Hugo Pfoertner, <a href="/A288574/b288574.txt">Table of n, a(n) for n = 0..10000</a>
%p A288574 := proc(n)
%p local a, i, j, k, p, q, r,pqr ;
%p a := 0 ;
%p for i from 2 do
%p p := ithprime(i) ;
%p for j from i do
%p q := ithprime(j) ;
%p for k from j do
%p r := ithprime(k) ;
%p if p+q+r = 2*n+1 then
%p pqr := {p,q,r} ;
%p a := a+nops(pqr) ;
%p elif p+q+r > 2*n+1 then
%p break;
%p end if;
%p end do:
%p if p+2*q > 2*n+1 then
%p break;
%p end if;
%p end do:
%p if 3*p > 2*n+1 then
%p break;
%p end if;
%p end do:
%p return a;
%p end proc:
%p seq(A288574(n),n=0..80) ; # _R. J. Mathar_, Jun 29 2017
%o (PARI) a(n)={my(p,q,r,cnt);n=2*n+1;
%o forprime(p=3,n\3,forprime(q=p,(n-p)\2,
%o if(isprime(r=n-p-q), cnt+=if(p===q&&p==r,1,if(p==q||q==r,2,3)))));cnt}
%o \\ _Franklin T. Adams-Watters_, Jun 28 2017
%o (Python)
%o from sympy import primerange, isprime
%o def a(n):
%o n=2*n + 1
%o c=0
%o for p in primerange(3, n//3 + 1):
%o for q in primerange(p, (n - p)//2 + 1):
%o r=n - p - q
%o if isprime(r): c+=1 if p==q and p==r else 2 if p==q or q==r else 3
%o return c
%o print([a(n) for n in range(66)]) # _Indranil Ghosh_, Jun 29 2017
%Y A288573 appears to be an erroneous version of this sequence.
%Y Cf. A007963, A054860, A087916.
%K nonn
%O 0,6
%A _Franklin T. Adams-Watters_ and _R. J. Mathar_, Jun 28 2017
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