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A332941 Lexicographically earliest sequence of positive numbers in which no set of consecutive terms sums to a prime.

%I A332941 #26 Jan 21 2023 19:43:28
%S A332941 1,8,1,15,9,1,14,6,30,6,9,15,6,4,8,12,10,14,6,12,8,10,12,18,12,6,6,6,
%T A332941 24,6,6,8,1,9,6,10,8,12,6,14,10,6,4,8,12,10,20,6,18,6,6,4,8,12,6,4,12,
%U A332941 8,10,8,6,6,18,6,6,20,10,12,8,4,6,12,12,6,12,6,12
%N A332941 Lexicographically earliest sequence of positive numbers in which no set of consecutive terms sums to a prime.
%C A332941 Terms >= 30 seem to be very rare. Up to a(450000), 30 appears only 7 times: at n = 9, 288, 2507, 15902, 54405, 242728, 425707.
%C A332941 For n <= 450000, the largest term is 32; it appears at n = 335308 and 370687.
%H A332941 Alois P. Heinz, <a href="/A332941/b332941.txt">Table of n, a(n) for n = 1..10000</a>
%p A332941 s:= proc(i, j) option remember; `if`(i>j, 0, a(j)+s(i, j-1)) end:
%p A332941 a:= proc(n) option remember; local k; for k while
%p A332941       ormap(isprime, [k+s(i, n-1)$i=1..n]) do od; k
%p A332941     end:
%p A332941 seq(a(n), n=1..100);  # _Alois P. Heinz_, Mar 23 2020
%t A332941 s[i_, j_] := s[i, j] = If[i > j, 0, a[j] + s[i, j-1]];
%t A332941 a[n_] := a[n] = Module[{k}, For[k = 1, AnyTrue[k+Table[s[i, n-1], {i, 1, n}], PrimeQ], k++]; k];
%t A332941 Array[a, 100] (* _Jean-François Alcover_, Nov 17 2020, after _Alois P. Heinz_ *)
%o A332941 (Python)
%o A332941 def A(ee):
%o A332941     a=[1]
%o A332941     print(1)
%o A332941     n=1
%o A332941     while n<=ee:
%o A332941         i=1
%o A332941         while i>0:
%o A332941             ii=i
%o A332941             iz=c=0
%o A332941             while iz<=len(a):
%o A332941                 c=0
%o A332941                 if ii>2:
%o A332941                     for j in range(2, int((ii)**0.5+1.5)):
%o A332941                         if ii%j==0:
%o A332941                             c=1
%o A332941                             break
%o A332941                 if c==0 and ii>1:
%o A332941                     break
%o A332941                 else:
%o A332941                     iz += 1
%o A332941                     ii=ii+a[n-iz]
%o A332941             if c==1:
%o A332941                 n += 1
%o A332941                 a.append(i)
%o A332941                 print(i)
%o A332941                 break
%o A332941             if i<4:
%o A332941                 i=4
%o A332941             else:
%o A332941                 i += 1
%o A332941     return a
%Y A332941 Cf. A254337, A084833.
%K A332941 nonn
%O A332941 1,2
%A A332941 _S. Brunner_, Mar 03 2020