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%I #70 Jan 27 2019 09:34:35
%S 2,3,5,7,97,101,521,787,907,911,1117,1151,1361,3251,4211
%N 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).
%C Conjecture 1: All prime numbers >= 11 can be written as sum of 3 twin prime numbers.
%C 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.
%C If the sequence is finite then there are infinitely many twin primes.
%C 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.
%C a(16) > 10^6 if it exists. - _Amiram Eldar_, Dec 06 2018
%H David A. Corneth, <a href="/A318202/a318202_1.gp.txt">PARI prog</a>
%e 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.
%t p = Prime[Range[600]]; p2 = Select[p, PrimeQ[# - 2] || PrimeQ[# + 2] &]; Select[ p - 3, IntegerPartitions[#, {2}, p2] == {} &] + 3 (* _Amiram Eldar_, Nov 15 2018 *)
%o (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", ")))}
%o (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
%o (PARI) \\ See Corneth link \\ _David A. Corneth_, Dec 05 2018
%Y Cf. A000040, A001097, A007534, A129363, A321221.
%K nonn,more
%O 1,1
%A _Dimitris Valianatos_, Aug 21 2018
%E 2,3,5,7 prepended by _David A. Corneth_, Dec 05 2018
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