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A068638 a(1) = 1, a(n) = smallest distinct composite number such that a(n) + a(k) is a composite number for all k = 1 to n.

%I A068638 #17 Dec 15 2022 09:59:59
%S A068638 1,8,14,20,24,25,26,32,38,44,50,56,62,68,74,80,86,90,92,94,98,104,110,
%T A068638 116,118,120,122,128,134,140,144,146,152,158,160,164,170,176,182,184,
%U A068638 188,194,200,206,212,218,220,224,230,234,236,242,248,254,260,264,266
%N A068638 a(1) = 1, a(n) = smallest distinct composite number such that a(n) + a(k) is a composite number for all k = 1 to n.
%C A068638 Conjecture: 25 is the largest odd term of this sequence.
%C A068638 Essentially the same as A025044. - _R. J. Mathar_, Sep 30 2008
%H A068638 Ray Chandler, <a href="/A068638/b068638.txt">Table of n, a(n) for n = 1..10000</a>
%e A068638 a(2) = 8 as for the smaller composite numbers 4 and 6 one gets 4 + 1 = 5 and 6 + 1 = 7, both primes. a(3) = 14 as 1 + 14 = 15 and 8 + 14 = 22 are composite.
%t A068638 a1 = {0}; nmax = 266; Do[ If[Select[n + a1, PrimeQ] == {}, AppendTo[a1, n]] , {n, nmax}]; Rest[a1] (* _Ray Chandler_, Jan 15 2017 *)
%o A068638 (Python)
%o A068638 from sympy import isprime
%o A068638 from itertools import islice
%o A068638 def agen(start=1): # generator of terms
%o A068638     alst, k, sums = [0, start], 2, {0} | {start}
%o A068638     while True:
%o A068638         yield alst[-1]
%o A068638         while any(isprime(k+an) for an in alst): k += 1
%o A068638         alst.append(k)
%o A068638         k += 1
%o A068638 print(list(islice(agen(), 60))) # _Michael S. Branicky_, Dec 15 2022
%Y A068638 Cf. A025044.
%K A068638 easy,nonn
%O A068638 1,2
%A A068638 _Amarnath Murthy_, Feb 27 2002
%E A068638 More terms from _Sascha Kurz_, Mar 17 2002
%E A068638 Description clarified by _Ray Chandler_, Jan 15 2017