%I #94 Jan 05 2020 22:22:20

%S 0,1,8,6,10,14,12,4,20,16,24,18,22,28,26,34,30,32,36,40,42,46,38,44,

%T 52,48,54,50,58,56,62,64,60,66,68,72,70,74,80,76,78,86,82,84,90,92,94,

%U 88,98,96,104,100,102,108,110,112,114,106,116,122,118,120,124,126,130,132,134,128,138,136,142,140,144,146,148,150,154,152,156,158

%N Lexicographically earliest sequence of distinct numbers such that no sum of consecutive terms is prime.

%C In other words, no sum a(i)+a(i+1)+a(i+2)+...+a(n) may be prime. In particular, the sequence may not contain any primes.

%C I conjecture that the sequence contains all even numbers > 2 and no odd number beyond 1. If so, we must simply ensure that the sum a(1)+...+a(n) is not prime, which is always possible for one of the three consecutive even numbers {2n, 2n+2, 2n+4}. As a consequence, it would follow that a(n) ~ 2n.

%C Is there even a proof that the smallest odd composite number, 9, does not appear?

%C The variant A254341 has the additional restriction of alternating parity, which avoids excluding the odd numbers.

%C The least odd composite number a'(n+1) that could occur as the next term after a(n) and such that sum(a(i),i=k...n)+a'(n+1) is composite for all k <= n is (for n = 0, 1, 2,...): 9, 9, 25, 21, 39, 25, 69, 65, 45, 119, 95, 77, 55, 27, 595, 561, 531, 865, 1519, 1479, 1437, 1391, 1353, 1309, 1257, 1209, 1155, 1105, 1047, 2317, 2255, 2191, 3565, 5719, 13067, 12995, 12925, 12851, 12771, 12695, 12617, 12531, 12449, 12365, 12275, ... The growth of this sequence shows how it is increasingly unlikely that an odd number could occur, since the next possible even term is only about 2n.

%H Robert G. Wilson v and M. F. Hasler, <a href="/A254337/b254337.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..999 from M. F. Hasler)

%F It appears that a(n) ~ 2n.

%e To explain the beginning of the sequence, observe that starting with the smallest possible terms 0, 1 does not appear to lead to a contradiction (and in fact never does), so we start there.

%e The next composite would be 4 but 1+4=5 is prime, as is 1+6, but 1+8=9 is not, so we take a(2) = 8 to be the next term.

%e 4 is impossible for a(3) since 1+8+4=13 is prime, but neither 1+8+6=15 nor 8+6 is prime, so a(3)=6.

%t f[lst_List] := Block[{k = 1}, While[ PrimeQ@ k || MemberQ[lst, k] || Union@ PrimeQ@ Accumulate@ Reverse@ Join[lst, {k}] != {False}, k++]; Append[lst, k]]; Nest[f, {0}, 70] (* _Robert G. Wilson v_, Jan 31 2015 *)

%o (PARI) a=[];u=0; for(i=1,99, a=concat(a,0); until( ! isprime(s) || ! a[i]++, while( isprime(a[i]) || bittest(u,a[i]), a[i]++); s=a[k=i]; while( k>1 && ! isprime( s+=a[k--]),)); u+=2^a[i]; print1(a[i]","))

%Y Cf. A254341, A153136, A254211, A002808, A253073, A253074, A054408, A084834.

%Y Cf. A025044 (no pairwise sum is prime), A025043 (no pairwise difference is prime).

%K nonn,nice

%O 0,3

%A _M. F. Hasler_, Jan 28 2015