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A318202
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Prime numbers p such that p-3 cannot be written as sum of two twin prime numbers (not necessarily forming a pair of twin primes).
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1
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2, 3, 5, 7, 97, 101, 521, 787, 907, 911, 1117, 1151, 1361, 3251, 4211
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OFFSET
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1,1
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COMMENTS
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Conjecture 1: All prime numbers >= 11 can be written as sum of 3 twin prime numbers.
Conjecture 2: All prime numbers aside from the 15 terms given here can be written as a sum of three twin prime numbers with (at least) one of them equal to 3.
If the sequence is finite then there are infinitely many twin primes.
For the terms in this sequence, the lexicographically least partitions into three twin primes are: 97=5+19+73, 101=11+17+73, 521=11+197+313, 787=5+139+643, 907=5+19+883, 911=11+17+883, 1117=5+19+1093, 1151=11+107+1033, 1361=11+29+1321, 3251=11+71+3169, 4211=11+41+4159.
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LINKS
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EXAMPLE
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a(6) = 101 because 101 - 3 = 98 and (98 - 73 = 25, 98 - 71 = 27), (98 - 61 = 37, 98 - 59 = 39), ..., (98 - 5 = 93, 98 - 3 = 95) aren't twin primes.
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MATHEMATICA
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p = Prime[Range[600]]; p2 = Select[p, PrimeQ[# - 2] || PrimeQ[# + 2] &]; Select[ p - 3, IntegerPartitions[#, {2}, p2] == {} &] + 3 (* Amiram Eldar, Nov 15 2018 *)
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PROG
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(PARI) {forprime(n=2, 10^4, p=n-3; forprime(t1=2, n, forprime(t2=t1, n, t12=t1+t2; if((isprime(t1-2)||isprime(t1+2))&&(isprime(t2-2)||isprime(t2+2)), if(t12==p, break(2))))); if(t12==2*n, print1(n", ")))}
(PARI) isok(p) = {if (isprime(p), p -= 3; forprime(q = 2, p, if (isprime(r=p-q), if ((isprime(r+2) || isprime(r-2)) && (isprime(q-2) || isprime(q+2)), return (0)); ); ); return (1)); } \\ Michel Marcus, Dec 05 2018
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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