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a(1) = 2. For n >= 2, a(n) = the smallest prime > a(n-1) + p(n) - p(n-1), where p(n) is the n-th prime.
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%I #13 Jan 22 2019 22:53:30

%S 2,5,11,17,23,29,37,41,47,59,67,79,89,97,103,113,127,131,139,149,157,

%T 167,173,181,191,197,211,223,227,233,251,257,269,277,293,307,317,331,

%U 337,347,359,367,379,383,389,397,419,433,439,443,449,457,461,479,487

%N a(1) = 2. For n >= 2, a(n) = the smallest prime > a(n-1) + p(n) - p(n-1), where p(n) is the n-th prime.

%C If instead we had: a(n) = the smallest prime >= a(n-1) + p(n) - p(n-1), then {a(k)} would simply be the sequence of primes, obviously.

%H Harvey P. Dale, <a href="/A156830/b156830.txt">Table of n, a(n) for n = 1..1000</a>

%p A156830 := proc(n) option remember; if n = 1 then 2; else nextprime(procname(n-1)+ithprime(n)-ithprime(n-1)) ; fi; end: seq(A156830(n),n=1..120) ; # _R. J. Mathar_, Feb 22 2009

%t nxt[{n_,a_}]:={n+1,NextPrime[a-Prime[n]+Prime[n+1]]}; NestList[nxt,{1,2},60][[All,2]] (* _Harvey P. Dale_, Sep 23 2016 *)

%Y A001223

%K nonn

%O 1,1

%A _Leroy Quet_, Feb 16 2009

%E Extended by _R. J. Mathar_, Feb 22 2009