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A302756
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a(n) is the least possible greatest prime in any partition of prime(n) into three primes; n >= 4.
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1
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3, 5, 5, 7, 7, 11, 11, 13, 13, 17, 17, 17, 19, 23, 23, 29, 29, 31, 31, 31, 31, 37, 41, 37, 41, 41, 41, 43, 47, 47, 53, 53, 61, 61, 61, 61, 61, 61, 61, 71, 67, 71, 71, 73, 79, 83, 79, 83, 83, 83, 89, 89, 97, 97, 101, 97, 101, 97, 103, 103, 107, 107, 107, 113, 127, 127
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OFFSET
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4,1
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COMMENTS
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Goldbach's weak (ternary) conjecture states that every odd number > 5 can be expressed as the sum of three primes (see link). This sequence applies the conjecture (now proven) to primes > 5. From all possible partitions of prime(n) = p+q+r for primes p,q,r (p <= q <= r), a(n) is chosen as the least possible value of the greatest prime r (with lower prime p not constrained to be A302607(prime(n)). The sequence is not strictly increasing, and although many primes appear repeatedly, some do not appear at all (e.g. 59 is not included).
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LINKS
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EXAMPLE
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The partition of prime(5)=11 into 3 primes p <= q <= r is 11=3+3+5 and since no smaller value than 5 can be attributed to r, a(5)=5.
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PROG
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(PARI) a(n) = {my(pn = prime(n), res = oo); forprime(p=2, pn, forprime(q=p, pn, forprime(r=q, pn, if (p+q+r == pn, res = min(res, r)); ); ); ); res; } \\ Michel Marcus, May 13 2018
(PARI) first(n) = {n = prime(n + 3); my(strt = vector(n, i, i), t = 0, res = vector(primepi(n) - 3)); forprime(p = 2, n, forprime(q = p, n - p, forprime(r = q, n - p - q, strt[p + q + r] = min(r, strt[p + q + r])))); forprime(p = 7, n, t++; res[t] = strt[p]); res} \\ David A. Corneth, May 14 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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