OFFSET
1,1
COMMENTS
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.
a(16) > 10^6 if it exists. - Amiram Eldar, Dec 06 2018
LINKS
David A. Corneth, PARI prog
EXAMPLE
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.
MATHEMATICA
p = Prime[Range[600]]; p2 = Select[p, PrimeQ[# - 2] || PrimeQ[# + 2] &]; Select[ p - 3, IntegerPartitions[#, {2}, p2] == {} &] + 3 (* Amiram Eldar, Nov 15 2018 *)
PROG
(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
(PARI) \\ See Corneth link \\ David A. Corneth, Dec 05 2018
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Dimitris Valianatos, Aug 21 2018
EXTENSIONS
2,3,5,7 prepended by David A. Corneth, Dec 05 2018
STATUS
approved