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If you sum 3 consecutive odd prime numbers p,q,r, you get a number s which is either prime or not: p+q+r=s. If s is prime, you call it p and repeat the game. If s is not prime, you call the largest prime factor p and repeat the game. Finally, you get into an infinite cycle, which is one of the above 3 sequences, no matter what initial prime numbers you choose.
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%I #7 Feb 18 2015 02:25:17

%S 7,31,109,349,1061,103,29,97,43,13,11,41,131,37,17,59

%N If you sum 3 consecutive odd prime numbers p,q,r, you get a number s which is either prime or not: p+q+r=s. If s is prime, you call it p and repeat the game. If s is not prime, you call the largest prime factor p and repeat the game. Finally, you get into an infinite cycle, which is one of the above 3 sequences, no matter what initial prime numbers you choose.

%C You can invent numerous variations which generate other cycles, but always you end in 2 or 3 cycles.

%F p+q+r=s, prime. s=p. repeat. p+q+r=s=f1*f2*f3..., fi prime. Largest f=p. repeat.

%e p=7

%e 7+11+13=31

%e 31+37+41=109

%e 109+113+127=349

%e 349+353+359=1061

%e 1061+1063+1069=3193=31*103

%e 103+107+109=319=11*29

%e 29+31+37=97

%e 97+101+103=301=7*43

%e 43+47+53=143=11*13

%e 13+17+19=49=7*7

%e 7+11+13...

%Y Cf. A117631.

%K fini,full,nonn

%O 1,1

%A _Werner D. Sand_, Aug 12 2005